Modern scholars tend to hold that Mark was the earliest gospel and that Matthew and Luke are, at least in part, using Mark as a source along with another, now lost, document (called "Q" from "Quelle", the German word for "source") and other materials which are known only to Matthew or Luke. The reason they think this is because of the profound similarities (often verbatim) between the three gospels. This so-called "two source" theory is not without its difficulties, however.

The first difficulty is the unanimity of ancient tradition concerning the primacy of Matthew’s gospel. Two source proponents would counter that it is quite possible Matthew composed a collection of "sayings of Jesus" in Aramaic first which was later incorporated into our present gospel or perhaps used by St. Mark when he wrote (whose gospel in turn was used as a source for our present version of Matthew). Others argue that it is unlikely an apostolic eyewitness would rely on the gospel of someone who was not an eyewitness.

Proponents of the two-source theory argue that since Mark was Peter’s scribe and was, in essence, preserving the essence of Peter’s preaching, it is quite possible that Matthew bowed to Petrine Primacy and made use of Peter’s materials rather than reinvent the wheel.

Critics of the two-source theory in turn counter that this would not explain why Matthew at times varies from Mark’s text in significant ways. Also, they argue that there is difficulty with attributing the gospel to Matthew using the two-source theory since the theory usually claims that it was written, not by Matthew but a "Matthean community" that was only giving what they remembered Matthew preaching. The difficulty with this is that the earliest witnesses we have to gospel are unanimous in attributing this to Matthew himself.

This is strengthened paradoxically by the fact that Matthew is an unlikely figure to which the anonymous gospel would be attributed if he were not, in fact, the author. There is nothing particularly special about him in the gospel narrative and he hails from a trade (tax collector) not likely to inspire confidence in a Jewish reader. The only conceivable reason the church would attribute the gospel to him, under such circumstances, is because he was, in fact, the author.

As you can see, the arguments are still ongoing and probably will be for some time. Therefore, the Church has not spoken definitively to the question of which gospel was composed first. However, the Church concurs definitively with the unanimous witness of the Christian Tradition that, whoever wrote first, the gospels were, according to the Second Vatican Council, written by "apostles and apostolic men" (Dei Verbum II, 7).