Yours is the kingdom
Whether the doxology belongs to the prayer as Jesus gave it or to the church's liturgy.
Matthew 6:9-13
9Pray like this: ‘Our Father in heaven, may your name be kept holy.
10Let your Kingdom come. Let your will be done on earth as it is in heaven.
11Give us today our daily bread.
12Forgive us our debts, as we also forgive our debtors.
13Bring us not into temptation, but deliver us from the evil one. For yours is the Kingdom, the power, and the glory forever. Amen.’
What each tradition says
Rabbinic / Midrashic
Kohler states the doxology added in Matthew, following a number of manuscripts, is a portion of 1 Chronicles 29:11. It was the liturgical chant with which the Lord’s Prayer was concluded in the Church, and it also occurs in Jewish ritual, where the whole verse is chanted at the opening of the Ark of the Law.
Kaufmann Kohler, The Jewish Encyclopedia (1905), art. "Lord's Prayer, The" — On the doxology at the end of the prayer (section 'Original Form and Meaning')
Patristic
Chrysostom treats the doxology as spoken by Christ: "For Thine is the kingdom, and the power, and the glory." He concludes that since the kingdom belongs to God, we should fear no one, for no one can withstand Him or divide His empire.
John Chrysostom, Homilies on the Gospel of Saint Matthew (Nicene and Post-Nicene Fathers, First Series, vol. 10; trans. G. Prevost) — Homily XIX, on Matt. vi. 13 (the doxology)
Catholic
Herbert Thurston states that the doxology "for Thine is the Kingdom," etc., which appears in the Greek textus receptus and was adopted in later editions of the Book of Common Prayer, is undoubtedly an interpolation.
Herbert Thurston, The Catholic Encyclopedia (New York: Robert Appleton Company, 1910), art. "Lord's Prayer" — On the doxology (origin of the text of the prayer)
Orthodox
According to Philaret, to better understand the Lord’s Prayer, it may be divided into the invocation, seven petitions, and the doxology.
Philaret (Metropolitan of Moscow), The Longer Catechism of the Orthodox, Catholic, Eastern Church — Q. 399-400 (the division of the Lord's Prayer)
Reformed / Protestant
Calvin finds it surprising that the clause "For thine is the kingdom" has been left out by the Latins. He notes that this clause agrees so well with the rest of the prayer.
John Calvin, Commentary on Matthew, Mark, Luke, Volume 1 (Calvin Translation Society) — On Matt. vi. 13 (the clause 'For thine is the kingdom')
Evangelical
Wesley explains “For thine is the kingdom” as God’s sovereign right over all created things. He explains “the power” as the executive power by which God governs all things in his everlasting kingdom. He explains “the glory” as the praise due from every creature for God’s power, wondrous works, and the mightiness of his kingdom, which endures forever.
John Wesley, John Wesley's Notes on the Bible — Notes on Matthew 6:13 (the doxology)