Thy kingdom come. Thy will be done in earth, as [it is] in heaven.
Note 30 at Mt 6:10: The word "kingdom" means "a political or territorial unit ruled by a sovereign." When applied to God, it could refer to all creation, since "his kingdom ruleth over all" (Ps 103:19), but "kingdom" more often applies to His rule in and through those who are submitted to Him. The parables of the kingdom in Mt 13 are clearly referring to the church. Compared with Lu 17:21, we see that "the kingdom of God" more specifically refers to Christ living and ruling in our hearts. So, praying "thy kingdom come" is praying for the expansion and influence of God's rule in the hearts of people everywhere and ultimately, the establishment of His physical kingdom here on earth at His second coming (Re 11:15 and 20:4).
The term "kingdom of heaven" is a term found thirty-three times in Matthew's Gospel. By comparing Mt 13:31 with Mr 4:30-31, it can be seen that the "kingdom of God" and the "kingdom of heaven" are identifying the same thing.
Note 31 at Mt 6:10: God's Word reveals (or is) His will (examples: 2Pe 3:9 and 3Jo 2). Praying for God's will means praying in agreement with His Word (compare with Ps 103:20 and 1Jo 5:14-15).
Note 32 at Mt 6:10: We do not have to live beggarly existences here on this earth and just wait till we all get to heaven to begin experiencing the benefits of our salvation. Jesus told us to pray "thy will be done in earth, as it is in heaven." In heaven, He will wipe away all tears from our eyes (Re 21:4), so here on earth we can pray and believe to receive "joy unspeakable and full of glory" (1Pe 1:8). Healing, prosperity, and all the other benefits of heaven are ours here on earth to the degree that we can believe and receive them.