Here are a few selections from the Catechism of the Catholic Church (CCC) on the Kingdom of God:
The kingdom of heaven was inaugurated on earth by Christ. This kingdom shone out before men in the word, in the works and in the presence of Christ". The Church is the seed and beginning of this kingdom. (CCC, 567)
-Its [The People of God’s] destiny, finally, “is the Kingdom of God which has been begun by God himself on earth and which must be further extended until it has been brought to perfection by him at the end of time.” (CCC, 782)
The Church is ultimately one, holy, catholic, and apostolic in her deepest and ultimate identity, because it is in her that “the Kingdom of heaven,” the “Reign of God,” already exists and will be fulfilled at the end of time. The kingdom has come in the person of Christ and grows mysteriously in the hearts of those incorporated into him, until its full eschatological manifestation. Then all those he has redeemed and made “holy and blameless before him in love,” will be gathered together as the one People of God, the “Bride of the Lamb,” “the holy city Jerusalem coming down out of heaven from God, having the glory of God.” (CCC, 865)
At the end of time, the Kingdom of God will come in its fullness. (CCC, 1060)