What evidence do we have that points to the fact that the gospels were
written by the apostles themselves? The first one who believed that they
were written by them was Irenaius in 180 AD.

It is impossible to prove absolutely that the apostles wrote the letters
of the New Testament. Actually, you should be aware that Luke, Acts
(written by Luke), James (brother of Jesus, not the apostle), Jude
(brother of Jesus), Mark (right hand man to Peter) and perhaps Hebrews
were not written by apostles. In fact, it is in general impossible to
prove conclusively that the writings of such famous ancient authors as
Herodotus, Cicero, Tacitus and so forth were actually written by these
men. The strongest evidence we have in general is that the contemporaries
accepted their writings as genuine. This is the case with the New
Testament. The strongest evidence we have that these writings were
apostolic is that they were accepted as such by the very earliest church
as a whole.

You are not correct to say that Irenaius was the first to claim that the
letters of Paul, Peter, John and the gospels were written by the apostles.
For example, one could mention the letter known as The Shepherd of Hermas,
as well as the letter of Clement of Rome to the Corinthians, both of which
were written in the last decade of the first century AD. Both letters
quote extensively from the letters of Paul, the gospels and the letters of
Peter, giving credit to the apostles for the books. The same could be said
for the letters of Ignatius of Antioch who wrote in the last decade of the
first and the first decade of the second century AD. Another important
witness is the letters of Polycarp. This early church father was a
personal acquaintance of the apostle John. One can assume he would know if
John had written the gospel of John. Polycarp quotes from the gospel of
John. A small manuscript known as the Muratorian fragment has been dated
to 180 AD. It mentions all the New Testament books except Matthew and Mark
(but it says Luke is the third gospel), Hebrews, James, 1 and 2 Peter and
1 John as being inspired and therefore having apostolic authority.
Irenaius was only quoting fairly long-standing tradition when he mentioned
the apostolic authority of the letters and the gospels at the end of the
second century AD.

In the end, we must rely on the authority of the letters themselves and
the witness of those in the early church who were clearly in a position to
judge whether these letters were apostolic by simply asking the apostles
whether they wrote them. Although there was some debate about the
apostolic authority for 2 and 3 John, Hebrews, and 2 Peter in the second
century, one can conclude with confidence that in nearly every case the
traditional author of the letters in the New Testament are those for whom
they are generally attributed. A couple of references for further reading
include The New Testament Documents: Are They Reliable? by F. F. Bruce and
How We Got the Bible, by Neil Lightfoot (see the books section of the web
site).

John Oakes, PhD