Scholarship on the Gospel of Matthew is on-going, just as one might expect. But some things are generally agreed upon.
There is general agreement on the time and place of this gospel's composition. Assuming Matthew's use of the Gospel of Mark and the hypothetical source called Q (from Quelle, German for "source") and its apparent awareness of the destruction of Jerusalem (22:7), which took place in A.D. 70, this gospel must have been composed in the last decades of the first century A.D. Moreover, since it appears that this gospel was quoted by Ignatius of Antioch and by the author of the Didache (both early second century), it must have been in existence some time before the turn of the century. The usual estimate is that it was composed somewhere about A.D. 80-90 in Syria (perhaps Antioch) or in nearby Palestine.
The Gospel of Matthew is an anonymous work, written by a Greek-speaking Jewish Christian who was acquainted with scribal habits of composition. The scribes of his day--and their rabbinic successors of the late first century and following--sought to preserve, systematize, and apply the traditions of their masters in order to fashion Jewish life for a new era following the destruction of the Temple. Eventually their work was preserved in written collections that contain the rabbinic traditions, particularly the Mishnah and the Talmuds of Babylonia and Jerusalem. So, too, Matthew sought to preserve, systematize, and apply the traditions of Jesus in order to fashion Christian community in his time and place. His gospel has often been called a handbook for parish life. If there is an author's "signature" within this gospel, it may well be at 13:52: "Every scribe who has been trained for the kingdom of heaven is like the master of a household who brings out of his treasure what is new and what is old." This seems to coincide with Matthew's own way of working. He preserves the old (often quoting from the Old Testament and using traditions about Jesus that he has received), and he applies them, addressing his readers with the message of Jesus concerning life together in community.
One of the most prominent features of this gospel is the way that the evangelist connects the ministry of Jesus with the Old Testament. He sees the Old Testament as being fulfilled on various occasions. Here he uses his famous "formula quotations," in which he says that a given event took place to fulfill what was written in the Old Testament, sometimes naming the book where the passage is found. The "formula quotations" appear at 1:22-23; 2:15, 17-18, 23; 4:14-16; 8:17; 12:17-21; 13:14-15, 35; 21:4-5; 27:9-10; cf. also 2:5-6.
Other features of the Gospel of Matthew are summarized in the essay called "Preaching from Matthew's Gospel: Major Themes and Forms of Teaching."
One of the matters on which interpreters disagree is how the Gospel of Matthew should be outlined. Some propose that it should be divided into three main portions: (1) 1:1-4:16 (on the identity of Jesus); (2) 4:17-16:20 (on Jesus' ministry); and (3) 16:21-28:30 (on Jesus' fate in suffering, death, and resurrection).
Others consider chapters 1-2 a preamble (section A below), chapters 26-28 an epilogue (section C), and chapters 3-25 a central section containing five blocks of narrative and discourse (section B). There is much to commend the latter. A feature that supports it in particular is that each of the five parts in section B ends with a recurring refrain ("When Jesus had finished saying these things" or the like). In this way, one can see how "Matthew the scribe" has systematized the teachings of Jesus in discourses on ethics, discipleship, parables, community discipline, and last things. The outline according to this view is as follows:
A. Birth and Infancy Narrative, 1:1-2:23
B. Central Section, 3:1-25:46
Part 1, 3:1-7:29
Narrative: Galilean Ministry (John, Baptism, Temptation, Call of First Disciples, et al.), 3:1-4:25
Discourse: Sermon on the Mount, 5:1-7:29
Concluding formula, 7:28: "Now when Jesus had finished saying these things, the crowds were astonished at his teaching, for he taught them as one having authority, and not as their scribes."
Part 2, 8:1-10:42
Narrative: Galilean Ministry (10 miracle stories), 8:1-9:38
Discourse: Instructing the Twelve on Discipleship and Mission, 10:1-42
Concluding formula, 11:1: "Now when Jesus had finished instructing his twelve disciples, he went on from there to teach and proclaim his message in their cities."
Part 3, 11:1-13:53
Narrative: Galilean Ministry (John, Woes on Cities, Controversies with Pharisees, Beelzebul Controversy, et al.), 11:1-12:50
Discourse: Seven Parables, 13:1-53
Concluding formula, 13:53: "When Jesus had finished these parables, he left that place."
Part 4, 13:54-18:35
Narrative: Galilean Ministry (Rejection, John's Death, Miracles, Peter's Confession, Transfiguration, Controversies with Pharisees, et al.), 13:54-17:27
Discourse: Community Discipline, 18:1-35
Concluding formula, 19:1: "When Jesus had finished saying these things, he left Galilee and went to the region of Judea beyond the Jordan."
Part 5, 19:1-25:46
Narrative: Ministry in Judea and Jerusalem (Controversies with Pharisees and Sadducees, Entry into Jerusalem, Cleansing of Temple, Woes upon Pharisees, Lament over Jerusalem, et al.), 19:1-23:39
Discourse: On Last Things, 24:1-25:46
Concluding formula, 26:1-2: "When Jesus had finished saying all these things, he said to his disciples, 'You know that after two days the Passover is coming, and the Son of Man will be handed over to be crucified.'"
C. Passion, Death, and Resurrection Narratives, 26:1-28:20
Regardless of how one outlines the Gospel of Matthew, it is evident that the teachings of Jesus are the main interest of the evangelist. Although it might be oversimplifying to do so, it is commonly said that while the evangelist who wrote the Gospel of Mark was especially interested in the activities of Jesus, the author of the Gospel of Matthew was particularly interested in his teaching. That feature makes the Gospel of Matthew an especially fruitful source for preaching.