Because there is still debate regarding the Synoptic Problem, the major solution theories will be considered below.
The Traditional Augustinian Theory:
This theory suggests that Matthew was the first Gospel to be composed, followed by Mark, then Luke. The second and third Gospels relied on the previous Gospel(s) as sources. Some view a preservation of Matthean priority as essential because of certain statements by early church fathers. One of those statements came from Augustine who said that the evangelists "have written in this order: first Matthew, then Mark, third Luke, and last John."1 Both Irenaeus and Origen, predecessors of Augustine, also held to the same order of composition.2 Later advocates of this theory include Hugo Grotius, H. G. Jameson, Basil Christopher Butler, and John Wenham.
The Two-Gospel Hypothesis (Griesbach Theory):
Like the Traditional Augustinian Theory, the Griesbach Theory also maintains Matthean priority. Unlike the previous theory, however, the Two-Gospel Hypothesis holds to Luke being the second Gospel, and then Mark as the third. Luke would have used Matthew as a source, and then Mark would have used both Matthew and Luke as sources. Again, Matthean priority finds support in the church fathers. It was Clement of Alexandria who wrote that the Gospels with genealogies (Matthew and Luke) were written first.3 Some of the main contributors to this theory are Henry Owen, J. J. Griesbach, William R. Farmer, and T. R. W. Longstaff et al. (http://www.colby.edu/rel/2gh/).
On the other hand, there are problems with Matthean priority. It is evident that Mark's Gospel is the shortest and the majority of it is also found in Matthew and Luke.4 It is difficult to explain why the shortest Gospel is only about ten percent original, especially when there is much support of it being Peter's interpretation through Mark.5 If indeed Mark was an abridgment of the Matthew and Luke, it also would be hard to give account for the deletion of significant points that are found in the two Gospels (e.g. Birth of Christ and the Sermon on the Mount). Moreover, many of the earliest quotes supporting Matthean priority also state that it was written in the Hebrew dialect (or Aramaic). Consequently, these quotes do not require Matthean priority in the Greek text, which will allow possibilities for Markan or Lukan priority.
Undoubtedly, this theory has become the most widely accepted theory amongst New Testament scholars today. The reason for its popularity is that it settles the problems that arise with Matthean priority, while confronting the difficulty of double tradition. Under the Two-Source Theory the priority is given to Mark. Both Matthew and Luke separately used Mark as a source. Matthew reproduces the vast majority of Mark, while Luke also incorporates more than half. In addition to using Mark as a source, Matthew and Luke also used a common source called Q.
The letter Q is short for the German word Quelle which means "source" or "spring." Q can actually refer to a few different things. It could be a tangible first century document, parts of various first century documents, oral tradition(s), or just the double tradition material that is found in both Matthew and Luke. Many differing hypotheses have been made concerning Q because there is no tangible proof that such a document existed outside of the double tradition. One of the only areas of consensus regarding Q is that it antedates both Matthew and Luke. Q would also be a Sayings-Gospel. Unlike the Gospels in the New Testament, Q would not contain narrative sections because the Q material in both Matthew and Luke are sometimes placed in different contexts. Q remains a hypothesis, though, and until there is weightier evidence, it is only one of the few solutions to the Synoptic Problem.
The less popular Three-Source Theory is very similar to the Two-Source Theory except in one aspect. Markan priority and the use of Q are both retained, but the difference between the two is that the Three-Source Theory holds to a Matthean influence on Luke. Thus, Mark was written first, then Matthew, and then Luke, with Matthew and Luke using the previous Gospel(s) as a source in addition to Q. Most scholars see this as improbable because the idea of Luke using Matthew seems to contradict the reasons for the development of Q.
This theory is based on and has all the elements of the Two-Source Theory. In addition to Matthew and Luke independently using Mark and Q, they each used material that was distinctive to themselves. The material that is exclusive to Matthew is called M tradition, and Luke's material is called L tradition. Since this is just a form of the Two-Source Theory, this is also heavily favored among scholars.
Farrer Theory (Mark without Q):
The last theory that will be discussed here is commonly called the Farrer Theory. Like the previous few theories, the Farrer Theory gives priority to Mark. Matthew was the second to be composed, followed by Luke. Matthew would have used Mark, while Luke would have used Mark and Matthew. This theory eliminates the need for a theoretical Q because both the triple tradition and the double tradition are explained without the need of an outside source. The leading supporters of this theory include J. H. Ropes, A. M. Farrer, M. D. Goulder, and Mark Goodacre (http://www.bham.ac.uk/theology/q/). The advantage of the Farrer Theory is that it seems to solve the Synoptic Problem without the use of hypothetical external documents.