The interpretation of the Four Living Creatures as the Four Evangelists is an ancient tradition in the Catholic Church.  The most common interpretation today is the one promulgated by Pope St. Gregory the Great from the sixth century.  However, St Irenaeus, writing in the second century has a different take.

I quote next from the Suburbanbanshee blog quoting Gregory's sermon on Ezekiel.

"For behold, it is said, “And as for the likeness of their faces: there was the face of a man, and the face of a lion on the right side of all four; and the face of an ox, on the left side of all four; and the face of an eagle on top of all four.” [Ezek. 1:10]

"‘For that these four winged living creatures would designate the four holy Evangelists, the’ beginnings ‘of each one of the Evangelic books testify. For because he began with [Christ's] human begetting, Matthew by right [is signified] by the human. Because of the crying in the desert, Mark is rightly designated by the lion. Because he commenced with sacrifice, Luke is well signified by the calf. Truly, because he began with the Divinity of the Word, John was fittingly signified by the eagle — he who stretched [upward], saying, “In the beginning was the Word, and the Word was with God, and the Word was God” [John 1:1] while… in the same substance of Divinity; as if he fixed his eyes on the Sun in the manner of an eagle."

Irenaeus in his work Against Heresies (Book III, Chapter 11, No. 8) offers an alternative (more ancient) viewpoint.

The first living creature was like a lion, Revelation 4:7 symbolizing His effectual working, His leadership, and royal power; the second [living creature] was like a calf, signifying [His] sacrificial and sacerdotal order; but the third had, as it were, the face as of a man,— an evident description of His advent as a human being; the fourth was like a flying eagle, pointing out the gift of the Spirit hovering with His wings over the Church. And therefore the Gospels are in accord with these things, among which Christ Jesus is seated. For that according to John relates His original, effectual, and glorious generation from the Father, thus declaring, In the beginning was the Word, and the Word was with God, and the Word was God. John 1:1 Also, all things were made by Him, and without Him was nothing made. For this reason, too, is that Gospel full of all confidence, for such is His person. But that according to Luke, taking up [His] priestly character, commenced with Zacharias the priest offering sacrifice to God. For now was made ready the fatted calf, about to be immolated for the finding again of the younger son. Matthew, again, relates His generation as a man, saying, The book of the generation of Jesus Christ, the son of David, the son of Abraham; and also, The birth of Jesus Christ was on this wise. This, then, is the Gospel of His humanity; for which reason it is, too, that [the character of] a humble and meek man is kept up through the whole Gospel. Mark, on the other hand, commences with [a reference to] the prophetical spirit coming down from on high to men, saying, The beginning of the Gospel of Jesus Christ, as it is written in Esaias the prophet,— pointing to the winged aspect of the Gospel; and on this account he made a compendious and cursory narrative, for such is the prophetical character. And the Word of God Himself used to converse with the ante-Mosaic patriarchs, in accordance with His divinity and glory; but for those under the law he instituted a sacerdotal and liturgical service. Afterwards, being made man for us, He sent the gift of the celestial Spirit over all the earth, protecting us with His wings. Such, then, as was the course followed by the Son of God, so was also the form of the living creatures; and such as was the form of the living creatures, so was also the character of the Gospel.

	IRENAEUS	GREGORY
MATTHEW	Face of a Man	Face of a Man (human)
MARK	Eagle	Lion
LUKE	Calf (Ox)	Calf (Ox)
JOHN	Lion	Eagle