In the New Testament, the word basileia can be translated by "kingship" (abstract noun), "kingdom" (concrete noun) or "reign" (action noun). The Kingdom of God lies ahead of us. It is brought near in the Word incarnate, it is proclaimed throughout the whole Gospel, and it has come in Christ's death and Resurrection. The Kingdom of God has been coming since the Last Supper and, in the Eucharist, it is in our midst. The kingdom will come in glory when Christ hands it over to his Father:
It may even be . . . That the Kingdom of God means Christ himself, whom we daily desire to come, and whose coming we wish to be manifested quickly to us. For as he is our resurrection, since in him we rise, so he can also be understood as the Kingdom of God, for in him we shall reign.
Source: Catechism of the Catholic Church (Vatican.va)