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Christians have different opinions on this.
The traditional view is that the Gospels were written by the people named - Matthew and John the Apostles, and Luke and Mark mentioned in the New Testament. That view is held by many Christians. Others view the Gospels as being compiled later and having the names attached to them, perhaps to reflect the sources they had come from or associating them with particular teachings in the early church.
The various Wikipedia articles will give more information about academic views on this (which are not necessarily Christian views).
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Our knowledge of who wrote the Gospels comes principally from two sources:
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Manuscripts
The manuscript evidence is unanimous.
A title would be found either in the superscript (the top) or the subscript (the bottom) of the manuscript, or both. There are manuscripts where these portions are missing and, as a result, the portion of the manuscript that survives has no title.
Focusing then on intact manuscripts (by which I mean manuscripts with a surviving superscript and/or subscript):
This is overwhelming evidence of authorship. If the authors were unknown, we should see a variety of names cropping up in the manuscript titles--this is exactly what we do not see.
The original audiences would have known who the author was; the idea that the document did not identify its author is speculative, it is supported by 0 manuscript evidence. But the idea that the recipients didn't know who wrote this document (keep in mind some of these recipients were willing to die for what these documents said) is implausible.
For a very deep dive reviewing the evidence on Gospel titles, see the work of Martin Hengel.
The manuscript evidence is 100% consistent in supporting the claim that the Gospels were written by Matthew, Mark, Luke, and John.
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Patristic writings
Numerous early Christian historians reported who wrote the Gospels--some of the most important of these historians are Irenaeus, Clement of Alexandria, Tertullian, Origen, Eusebius, Epiphanius, and Jerome. Among these heavyweight scholars it is extraordinarily rare to find unanimity--they disagree about almost everything. But on this specific topic--who wrote the Gospels--their testimonies are in agreement. All of these scholars (and many more who could be named) reported Matthew, Mark, Luke, and John as the authors of the Gospels.
The scholars I’ve named represent multiple generations and tremendous geographic diversity--they cover the breadth of the Mediterranean: Lyons, Alexandria (x2), Carthage, Caesarea, Salamis, and Dalmatia.
Let’s consider very briefly some highlights of the evidence for each of the Gospels:
Matthew
Matthew is far and away the most quoted Gospel in the early church--it was known and trusted from the earliest patristic writings. It is quoted by Ignatius & Polycarp. Eduard Masseux argued effectively that Matthew is also quoted in 1 Clement, given exact correspondence in unusual Greek wording. (see pp. 21-24 here) Matthew is also quoted by the Didache, which may have been written in the 1st century. A passage from Matthew is found in the Epistle of Barnabas; the date of this epistle is uncertain, but Robinson (Redating the New Testament Ch. 10) makes a compelling argument that it was written in the 1st Century.
Matthew is mentioned as an author by name by Papais (~105), and explicitly as the author of the first Gospel by Irenaeus (~180), but it is quoted as a known, accepted, authoritative source as early as the first century--by people who knew the apostles personally.
The absence of any controversy over authorship--not the slightest insinuation even among the enemies of Christianity, is telling evidence that the third-generation Christians did not try to rename the Gospel of Matthew. Therefore, the author to whom the first Gospel was attributed by Irenaeus is the same person to whom the Gospel was attributed by the students of the apostles (second-generation Christians like Clement, Ignatius, Polycarp)
Mark
Papias of Hieropolis preserved in his history (written ~105) the testimony of a first-generation Christian Elder, indicating that Mark wrote a Gospel based on the preaching of Peter. Clement of Alexandria recorded this information a century later as well. Since Clement provides a number of details not found in any known fragment of Papias (and since Clement was from the Alexandrian church Mark is said to have founded), it is likely that Clement has at least some independent information (For the testimony of Papias, see HE 3.39; for several of the key statements by Clement see HE 2.15 & HE 6.14).
Two very well-informed scholars report that Mark wrote the second Gospel, they corroborate each other, their testimony is sustained by every Patristic writer after them, and there are exactly 0 competing claims.
Why is the Gospel of Mark multiply attested (by early sources) as having been written by such an obscure figure? (this is not the story you’d tell if you were making it up!) Far and away the simplest explanation is that it is true.
Luke
Our earliest surviving source naming Luke as the author appears to be the Muratorian Canon, written approx. AD 170; Irenaeus also names Luke as the author when writing circa AD 180. However, we have much earlier evidence that the Gospel of Luke was written by a disciple of Paul--we just wouldn’t know which one without the 2nd century sources.
Luke-Acts forms a two-part work by the same author. Colin Hemer has masterfully argued that the Book of Acts was written by an eyewitness disciple of Paul circa AD 62 (see here) (see also an excellent summary by Frank Luke on SE-Biblical Hermeneutics here). The author is intimately familiar with the people, time, and places he describes.
We can therefore show deductively that Luke was written by a disciple of Paul:
Courtesy of the Muratorian Canon and the writings of Irenaeus (to say nothing of the numerous later sources corroborating the claim), we know which disciple of Paul. But that it was an eyewitness disciple of Paul can be established from first-century sources.
John
Irenaeus of Lyons is arguably our most important external witness to Johannine authorship. He was from the part of the world where John lived in his later years (Irenaeus only moved to Gaul later in life), and was a pupil of Polycarp of Smyrna, who was a disciple of John. Irenaeus is just one link removed from apostolic testimony and, crucially, only one link removed from the author he claims wrote the Gospel of John.
Irenaeus grew up in a world saturated with John’s influence. He studied the works of Papias (another disciple of John). If Irenaeus believed John wrote John, that is enormously important historical evidence.
So did Irenaeus believe John wrote John or was he just saying that to further an agenda?
Irenaeus has often been misrepresented by those who do not appreciate his method of argumentation. Irenaeus argues from premises to conclusions. The premises are things that are generally known, the conclusions are the things he wants to prove. His conclusions could be garbage (I think some of his conclusions are clearly not garbage, but certainly some of his conclusions do appear to be false), but that doesn’t matter for our analysis here. We really don’t care about the process of reasoning Irenaeus used to get from his premises to his conclusions—because his attestation of Johannine authorship is used as a premise!
It may take a minute for the impact of that last statement to sink in. Irenaeus uses John’s authorship of the 4th Gospel as a premise (see Against Heresies Book 3 chapter 1). This means in his day and age (circa 180), John’s authorship of the 4th Gospel was generally known. Irenaeus is tearing apart (no joke, read Against Heresies) his opponents’ beliefs; if he cites an obviously false premise, he leaves himself open to a devastating counter-argument.
That Irenaeus can get away with baldly stating that John wrote John, without having to argue for it, in a world just 1 generation removed from John, indicates that this was a statement that was nearly incontrovertible at that time.
For a quick but scholarly review of the evidence of Johannine authorship, Erik Manning covers both the internal and external evidence.
There is a thread on SE-Biblical Hermeneutics suggesting John could not have written the Gospel of John--my thoughts in counter to that claim are here.
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Patristic evidence for authorship by Matthew & Mark is early and unanimous
For Luke the evidence for authorship by an eyewitness disciple of Paul is exceptionally strong. That the disciple's name was Luke is well-attested, but not as well as that of Matthew & Mark.
For John, the evidence is almost unanimous, and it comes only one link removed from John himself.
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Conclusion
The historical evidence is remarkably solid--yet, as many are aware, critical scholarship in the last 2 centuries has rejected most of this evidence. F. David Farnell put it very well:
Could it be that Enlightenment-spawned Historical Criticism has so systematically ignored the early fathers because they stand as manifest contradictions to its cherished dogmas?
There are implications to Gospel authorship, and some do not like those implications. But the evidence itself is robust - authorship of the Gospels by Matthew, Mark, Luke, and John is very well-attested. Introducing later redactors (for whom we do not have evidence) to explain the authorship of the Gospels usually requires multiplying entities beyond necessity.
I am in the process of producing a video series on Gospel authorship--those interested can find it here. I find that the evidence supports the view that the Gospels were indeed written by Matthew, Mark, Luke, and John.