I did a search for the word "pray," using the Bible Gateway, and the King James Version (KJV) of the Bible. The KJV uses outdated language, but is in the public domain, so that I can include it on this page without violating copyright. A search for the word, "pray," without specifying whole words only, also returns such words as "prayer" and "praying." I searched the New Testament from Romans through Revelation, because I supposed that these 22 books give the best possible picture of the New Testament church. There were 59 occurrences of "pray," or a word containing it, in these books.
Perhaps the greatest mass conversion of unbelievers in history took place in conjunction with Pentecost. Although it is possible that the 120 disciples, meeting as described in Acts 1-2, were praying for non-believers, there's no evidence of that. It's hard to believe that they were praying for each of the 3,000 or so who came to belief, since this was a diverse group, people from several different racial and ethnic types, even speaking languages that the 120 didn't know. But the 120, especially Peter, were transformed, as a result of drawing aside and praying, and as a result of the power of the Holy Spirit. Jesus had told them to wait for the "promise of the Father," the infilling of the Holy Spirit, so presumably that's what they were mostly praying for.
I suppose that this should be prayer for God's will for the life of the individual I am praying for, and also prayer for God's will generally--where does God want to work at this time, and how?
In the letters of the New Testament, the readers were asked, and expected, to pray for Christians who were leaders. Here are some examples:
Romans 15:30: Now I beseech you, brethren, for the Lord Jesus Christ's sake, and for the love of the Spirit, that ye strive together with me in your prayers to God for me;
2 Corinthians 1:11: Ye also helping together by prayer for us, that for the gift bestowed upon us by the means of many persons thanks may be given by many on our behalf.
Colossians 4:3: Withal praying also for us, that God would open unto us a door of utterance, to speak the mystery of Christ, for which I am also in bonds:
1 Thessalonians 5:25: Brethren, pray for us.
Hebrews 13:18: Pray for us: for we trust we have a good conscience, in all things willing to live honestly.
And then, of course, we have the best example possible, namely Jesus Christ. One of His most important prayers is recorded in John 17. The heart of that prayer is verses 9 - 18:
10 And all mine are thine, and thine are mine; and I am glorified in them.
11 And now I am no more in the world, but these are in the world, and I come to thee. Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are.
12 While I was with them in the world, I kept them in thy name: those that thou gavest me I have kept, and none of them is lost, but the son of perdition; that the scripture might be fulfilled.
13 And now come I to thee; and these things I speak in the world, that they might have my joy fulfilled in themselves.
14 I have given them thy word; and the world hath hated them, because they are not of the world, even as I am not of the world.
15 I pray not that thou shouldest take them out of the world, but that thou shouldest keep them from the evil.
16 They are not of the world, even as I am not of the world.
17 Sanctify them through thy truth: thy word is truth.
18 As thou hast sent me into the world, even so have I also sent them into the world.
Christ's prayer was that Christians will be sent out into the needy world around us, and kept from the evil in that world.
Clearly, praying for other Christians is very important. I wish that I had done more of it, and that other Christians had prayed more for me. What should I pray for other Christians? What I should pray for them is that they become, or remain, pure in an impure world, and that their ministry be effective. I believe that these are the main things that the New Testament teaches that Christians ought to ask God for. (I should also thank God for them in my prayers.) This does not rule out praying for my own needs, and those of others, be they believers or not, but, it seems, if we place most of our prayer efforts on praying for other Christians to be pure and effective, God's kingdom can advance as it should, and many of the other things will take care of themselves.
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My wife found a typo on July 25th, which I corrected. Thanks, and sorry.
I have also posted a list of places in Acts where a group of the members of the early church prayed together.
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On October 20, 2010, I found a typographical error, and corrected it.
I also noted, in reviewing the scripture used in this post, that there doesn't seem to be an example of prayer that Christians will have sufficient funds for their ministry.
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On March 7, 2013, I added the fifth paragraph, on Pentecost.
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On May 3, 2014, I posted this related bible study, on things that the New Testament says are in God's will, so that we can pray for them with confidence.
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On July 25, 2018, I re-did the search for the word, "prayer," and variants thereof, using the Blue Letter Bible, and the New International Version New Testament. This returned 72 uses of "pray" in the NIV NT, in 68 verses, plus 32 of "prayer," 21 of "prayed," 24 of "praying," and 3 of "prays." My search showed me that only 1 John 1:2, Acts 28:8, and James 5:14-16 mention praying for the sick.