The Kingdom of God is the rule and governance of the Godhead over all things. The Bible speaks of it existing in the past (Daniel 4:17, 25, 34, 5:21), present (Matthew 12:28, Luke 17:20 - 21, Colossians 1:13), and future (Daniel 2:44 - 45, 7:13 - 14, Revelation 11:15 - 18, 20:4 - 15, 21:1 - 22:5).
The phrase "Kingdom of Heaven" is found 33 times in 32 King James verses. Interestingly, all these occurrences are recorded only in the book of Matthew. The only verse in the Bible that uses the phrase twice quotes Jesus' words during his famous Sermon on the Mount. The Lord, while teaching the multitudes, makes it clear how important it is to obey God.
Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven (Matthew 5:19, KJV).
The synonymous phrase "Kingdom of God" is recorded 70 times in 69 verses. This time it is the book of Luke that uses it twice in one verse (Luke 17:20). He also writes it the most with 33 occurrences followed by Mark with 15. Even Matthew brought some variety to his references by utilizing it five times in the KJV (Matthew 6:33, 12:28, 19:24, 21:31 and 21:43). His first use was his take on the Lord's admonishment on what our life priorities should be.
But seek ye first the kingdom of God, and his righteousness: and all these things shall be added unto you (Matthew 6:33, KJV).
How active is it?
It should be noted that although the Godhead rules the universe, it is not actively governing the affairs of man. If it were, then there would be no need for Christ's climatic Second Coming and his millennial rule, from Jerusalem, over all the earth! Individual believers today, however, are being ruled and trained in holiness, on a one-to-one basis, by the Holy Spirit within them.
God is patient, kind, loving and longsuffering with humans. He will do whatever it takes to convince every human that His way is the best and ONLY way to exist for all eternity. There is, however, a time when all humans will have to decide whether to base their existence on selfless love or selfish lusts.
Only those who willingly submit to and obey God (Revelation 22:14), who wholeheartedly desire to live his way for all eternity, will be allowed entrance into eternal life. They will be sealed righteous forever and have access to the tree of life (Revelation 22:11, 14).
Those who also have the truth revealed to them but still adamantly reject their Creator will also have their choice honored. The Godhead will ultimately not force, possess or otherwise make anyone, against his or her will, repent and obey. Those who stubbornly reject righteousness and doggedly desire to live selfishly will not be allowed to enter the Kingdom of God. They, instead, will be removed from existence through the lake of fire (Revelation 21:8).