The Greek word behind kingdom, like the Hebrew and Aramaic words behind it, names the exercise of rule rather than the territory ruled. God reigning is the subject, and a great deal of misreading follows from taking the phrase as a place, so that the kingdom becomes a synonym for heaven understood as a destination after death. Matthew habit of writing kingdom of heaven where the other gospels write kingdom of God does not help, and is generally explained as a Jewish reticence about the divine name rather than as a different subject.
The parables describe it in terms of growth and hiddenness. It is a mustard seed that becomes a shrub birds nest in, leaven worked invisibly through a large quantity of dough, a treasure found in a field that is worth everything a man has. The consistent note is disproportion between a small and unpromising beginning and an outcome nobody would have predicted from it.
One saying is famously ambiguous in translation. Asked when the kingdom would come, Jesus replies in Luke that it is not coming with things that can be observed, and that it is entos hymon, which can be rendered within you or among you or in your midst. The choice between them changes the sentence from an inward and individual claim to a claim about a reality already present in the group being addressed, and translators have gone both ways for centuries.