The Lord's Prayer is full of many individual prayers and petitions, each with a variety of meaning.
For example, the phrase "Thy kingdom come" can refer to several spiritual themes.
The Catechism of the Catholic Church lays out the depth of this prayer in its opening paragraph on this petition:
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.
More specifically, the Kingdom of God has already come to us in the Eucharist:
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
In this sense, the Kingdom of God is already here and we participate in it whenever we go to Mass.
Final coming
At the same time, we are also praying for God's kingdom to come at the end of time:
In the Lord's Prayer, "thy kingdom come" refers primarily to the final coming of the reign of God through Christ's return. But, far from distracting the Church from her mission in this present world, this desire commits her to it all the more strongly. Since Pentecost, the coming of that Reign is the work of the Spirit of the Lord who "complete[s] his work on earth and brings us the fullness of grace."
While we may be asking for Jesus to come again and for this world to pass away, we are also beseeching Christ to work now in this world.
This is an important distinction to make, as it could be tempting to simply pray for Jesus' coming and to do nothing. We still have obligations in this world and we need to cooperate with God's grace to do good in society:
By a discernment according to the Spirit, Christians have to distinguish between the growth of the Reign of God and the progress of the culture and society in which they are involved. This distinction is not a separation. Man's vocation to eternal life does not suppress, but actually reinforces, his duty to put into action in this world the energies and means received from the Creator to serve justice and peace.
While "Thy kingdom come" may seem like a simple petition, it is very complex and includes many layers of meaning.