Short answer
A great deal. Both say there is one God, who created the heavens and the earth. Both believe in angels, prophets and revealed scripture. Both honour Jesus, born of the virgin Mary, and both expect him to return. Both teach that people will answer for their deeds on a Day of Judgement. Both teach that God (﷾) forgives those who forgive others. Both pray, and both fast. The moral law is largely the same: most of the Ten Commandments, much the same forbidden marriages, and no drunkenness. The Qur'an says that those who say "We are Christians" are the nearest in love to the believers. Where Islam stands apart: God (﷾) is One alone, and Jesus was a mighty prophet and a servant of God (﷾). Muslims explain it simply: both messages came from the same God (﷾). Where they part on belief, it is because the Bible was later changed or misread.
Full explanation
One God (﷾), the Creator
The Law of Moses declares that God (﷾) is one, and Jesus called this the first of all the commandments. The Qur'an says the same in Surat al-Ikhlas, the short chapter on God's (﷾) oneness. It invites Christians to come to common terms: to worship none but God (﷾) and to set up no partners with Him.
“Hear, O Israel: The Lord our God is one Lord:”
“And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:”
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
“Say: "O People of the Book! come to common terms as between us and you: That we worship none but Allah; that we associate no partners with him; that we erect not, from among ourselves, Lords and patrons other than Allah." If then they turn back, say ye: "Bear witness that we (at least) are Muslims (bowing to Allah's Will).”
The same story of creation
Both books say God (﷾) created the heavens and the earth in six days. Both describe man formed from the earth, with God's (﷾) breath or spirit breathed into him. The Qur'an adds that no weariness touched God (﷾), where Genesis says He rested.
“In the beginning God created the heaven and the earth.”
“For in six days the Lord made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the Lord blessed the sabbath day, and hallowed it.”
“We created the heavens and the earth and all between them in Six Days, nor did any sense of weariness touch Us.”
“And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made.”
“And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.”
“Behold! thy Lord said to the angels: "I am about to create man, from sounding clay from mud moulded into shape; "When I have fashioned him (in due proportion) and breathed into him of My spirit, fall ye down in obeisance unto him."”
The Ten Commandments
The Ten Commandments are the ten laws God (﷾) gave to Moses. The Qur'an repeats the heart of them. Worship God (﷾) alone; honour your parents; do not kill; do not commit adultery; do not steal. Surat al-Isra gives several of them in one passage. The pledge in Surat al-Mumtahana repeats several of them again.
“Thou shalt have no other gods before me.”
“Thy Lord hath decreed that ye worship none but Him, and that ye be kind to parents. Whether one or both of them attain old age in thy life, say not to them a word of contempt, nor repel them, but address them in terms of honour. And, out of kindness, lower to them the wing of humility, and say: "My Lord! bestow on them thy Mercy even as they cherished me in childhood."”
“Honour thy father and thy mother: that thy days may be long upon the land which the Lord thy God giveth thee.”
“Thou shalt not kill.”
“Nor take life - which Allah has made sacred - except for just cause. And if anyone is slain wrongfully, we have given his heir authority (to demand qisas or to forgive): but let him not exceed bounds in the matter of taking life; for he is helped (by the Law).”
“Thou shalt not commit adultery.”
“Nor come nigh to adultery: for it is a shameful (deed) and an evil, opening the road (to other evils).”
“Thou shalt not steal.”
“O Prophet! When believing women come to thee to take the oath of fealty to thee, that they will not associate in worship any other thing whatever with Allah, that they will not steal, that they will not commit adultery (or fornication), that they will not kill their children, that they will not utter slander, intentionally forging falsehood, and that they will not disobey thee in any just matter,- then do thou receive their fealty, and pray to Allah for the forgiveness (of their sins): for Allah is Oft-Forgiving, Most Merciful.”
Forbidden marriages and drink
The Law of Moses and the Qur'an forbid marriage within the close family: mothers, sisters, a father's wife and others. The New Testament warns against drunkenness. The Qur'an forbids intoxicants outright.
“None of you shall approach to any that is near of kin to him, to uncover their nakedness: I am the Lord. The nakedness of thy father, or the nakedness of thy mother, shalt thou not uncover: she is thy mother; thou shalt not uncover her nakedness. The nakedness of thy father’s wife shalt thou not uncover: it is thy father’s nakedness. The nakedness of thy sister, the daughter of thy father, or daughter of thy mother, whether she be born at home, or born abroad, even their nakedness thou shalt not uncover.”
“And marry not women whom your fathers married,- except what is past: It was shameful and odious,- an abominable custom indeed.”
“Prohibited to you (For marriage) are:- Your mothers, daughters, sisters; father's sisters, Mother's sisters; brother's daughters, sister's daughters; foster-mothers (Who gave you suck), foster-sisters; your wives' mothers; your step-daughters under your guardianship, born of your wives to whom ye have gone in,- no prohibition if ye have not gone in;- (Those who have been) wives of your sons proceeding from your loins; and two sisters in wedlock at one and the same time, except for what is past; for Allah is Oft-forgiving, Most Merciful;-”
“And be not drunk with wine, wherein is excess; but be filled with the Spirit;”
“O ye who believe! Intoxicants and gambling, (dedication of) stones, and (divination by) arrows, are an abomination,- of Satan's handwork: eschew such (abomination), that ye may prosper.”
Jesus, his virgin birth and his return
Luke's Gospel, one of the four New Testament Gospels, and the Qur'an both tell of the angel's message to Mary. Both give her answer that no man had touched her. Both say Jesus was taken up to God (﷾). Both say he will come again. The angels in Acts, the New Testament book about the early church, say so. The Prophet (ﷺ) swore that the son of Mary will descend as a just ruler.
“Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
“Behold! Allah said: "O Jesus! I will take thee and raise thee to Myself and clear thee (of the falsehoods) of those who blaspheme; I will make those who follow thee superior to those who reject faith, to the Day of Resurrection: Then shall ye all return unto me, and I will judge between you of the matters wherein ye dispute.”
“Which also said, Ye men of Galilee, why stand ye gazing up into heaven? this same Jesus, which is taken up from you into heaven, shall so come in like manner as ye have seen him go into heaven.”
Abu Huraira reports that the Prophet (ﷺ) swore that the son of Mary will soon descend among the Muslims as a just ruler. He will break the cross, kill the swine and end the jizya, and wealth will be so plentiful that no one will accept it. Abu Huraira then recited Qur'an 4:159: none of the People of the Book but will believe in him before his death.
Prayer and forgiveness
The prayer Jesus taught and the opening chapter of the Qur'an both praise God (﷾), submit to His will, and ask for guidance and help. Jesus taught his disciples to ask God (﷾) directly for forgiveness. He then explained that God (﷾) forgives those who forgive others. The Qur'an says the same. No sacrifice has to be paid first. The closing words, "For thine is the kingdom...", are missing from the earliest Greek manuscripts of Matthew.
“After this manner therefore pray ye: Our Father which art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done in earth, as it is in heaven. Give us this day our daily bread. And forgive us our debts, as we forgive our debtors. And lead us not into temptation, but deliver us from evil: For thine is the kingdom, and the power, and the glory, for ever. Amen.”
“In the name of Allah, Most Gracious, Most Merciful. Praise be to Allah, the Cherisher and Sustainer of the worlds; Most Gracious, Most Merciful; Master of the Day of Judgment. Thee do we worship, and Thine aid we seek. Show us the straight way, The way of those on whom Thou hast bestowed Thy Grace, those whose (portion) is not wrath, and who go not astray.”
“For if ye forgive men their trespasses, your heavenly Father will also forgive you: But if ye forgive not men their trespasses, neither will your Father forgive your trespasses.”
“Let not those among you who are endued with grace and amplitude of means resolve by oath against helping their kinsmen, those in want, and those who have left their homes in Allah's cause: let them forgive and overlook, do you not wish that Allah should forgive you? For Allah is Oft-Forgiving, Most Merciful.”
Fasting
Jesus taught his followers how to fast: in secret, for God (﷾) and not to be seen by people. The Qur'an prescribes fasting for the believers. It says fasting was prescribed for those before them too.
“Moreover when ye fast, be not, as the hypocrites, of a sad countenance: for they disfigure their faces, that they may appear unto men to fast. Verily I say unto you, They have their reward. But thou, when thou fastest, anoint thine head, and wash thy face; That thou appear not unto men to fast, but unto thy Father which is in secret: and thy Father, which seeth in secret, shall reward thee openly.”
“O ye who believe! Fasting is prescribed to you as it was prescribed to those before you, that ye may (learn) self-restraint,-”
The Day of Judgement
Both books teach that every person will be judged for what they did. Jesus says the wicked go away into everlasting punishment and the righteous into eternal life. The Qur'an says that anyone who has done even an atom's weight of good or evil will see it.
“And these shall go away into everlasting punishment: but the righteous into life eternal.”
“Then shall anyone who has done an atom's weight of good, see it! And anyone who has done an atom's weight of evil, shall see it.”
Nearest in love
The Qur'an says that of all people, those nearest in love to the believers are those who say "We are Christians". Among them are devoted scholars and monks, and they are not arrogant. The next verse describes them weeping when they hear the Qur'an, because they recognise the truth. Muslims believe in the Torah given to Moses and the Injil given to Jesus, as well as the Qur'an given to Muhammad (ﷺ).
“nearest among them in love to the believers wilt thou find those who say, "We are Christians": because amongst these are men devoted to learning and men who have renounced the world, and they are not arrogant.”
“Say ye: "We believe in Allah, and the revelation given to us, and to Abraham, Isma'il, Isaac, Jacob, and the Tribes, and that given to Moses and Jesus, and that given to (all) prophets from their Lord: We make no difference between one and another of them: And we bow to Allah (in Islam)."”
Where Islam stands apart
Islam holds that God (﷾) is One alone. Surat al-Ikhlas says that He begets not, nor is He begotten. The Qur'an honours Jesus as a messenger of Allah (﷾). It records him saying that he is a servant of Allah (﷾) and a prophet. Here Islam holds to the oneness of God (﷾) that Jesus himself taught.
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
“O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.”
“He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet;”
Perspectives
Mainstream Christian
Christians share with Muslims belief in one Creator, the prophets, the moral law and the judgement. For Christians, though, the heart of the faith is that Jesus is the Son of God, and that his death and resurrection bring salvation.
Luke says that the child born of the virgin "shall be called the Son of God".
Rebuttal
Islam agrees with Luke that the birth was God's (﷾) miracle, but the Bible also calls Adam and Israel God's son. A birth without a father cannot make Jesus God's Son: the Qur'an likens him to Adam, whom God (﷾) created from dust (3:59). It adds that it is not befitting God (﷾) to beget a son (19:35).
The Islamic perspective
Islam and Christianity agree on so much because both came from the same God (﷾) through His prophets. Where they part, it is on the nature of Jesus and on what Christians believe about the cross. There the Bible was changed or misread. The Qur'an restores the original message that Moses and Jesus brought.
The Law and the Qur'an both declare that God (﷾) is one. Jesus called this the first of all the commandments.
The Ten Commandments and the forbidden degrees of marriage in the Torah are largely matched in the Qur'an.
The Qur'an affirms the virgin birth of Jesus, and the Prophet (ﷺ) swore that he will return.
The Qur'an says that those who say "We are Christians" are the nearest in love to the believers. It invites the People of the Book to common terms: worship of God (﷾) alone.
The agreement comes from a shared source. The Qur'an confirms what was revealed to Moses and Jesus, and corrects what was changed.