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Best understood as the kingship, or sovereign and saving rule, of Israel's God YHWH, as celebrated in several Psalms (e.g. Psalm 99.1) and prophecies (e.g. Daniel 6.26-7). Because YHWH was the creator God, when he finally became king in the way he intended this would involve setting the world to right, and particularly rescuing Israel from its enemies. 'Kingdom of God' and various equivalents (e.g. 'No king but God!') became revolutionary slogans around the time of Jesus. Jesus' own announcement of God's kingdom redefined these expectations around his own very different plan and vocation. His invitation to people to 'enter' the kingdom was a way of summoning them to allegiance to himself and his programme, seen as the start of God's long-awaited saving reign. For Jesus, the kingdom was coming not in a single move, but in stages, of which his own public career was one, his death and resurrection another, and a still future consummation another. Note that 'kingdom of heaven' is Matthew's preferred form of the same phrase, following a regular Jewish practice of saying 'heaven' rather than 'God'. It does not refer to a place ('heaven'), but to the fact of God's becoming king in and through Jesus and his achievement. Paul speaks of Jesus, as Messiah, already in possession of his kingdom, waiting to hand it over finally to the father (1 Corinthians 15:23-8; cf. Ephesians 5.5).
N.T. Wright (Matthew For Everyone Part Two, pg. 218) (Not an endorsement for everything by NT Wright)