There is a lot of debate over the translation and meaning of 1 Corinthians 7:36-38. The NET Bible gives a fairly neutral reading of the passage, and the translators notes are helpful in summarising the arguments:
7:36 If anyone thinks he is acting inappropriately toward his virgin,(26) if she is past the bloom of youth(27) and it seems necessary, he should do what he wishes; he does not sin. Let them marry.
7:37 But the man who is firm in his commitment, and is under no necessity but has control over his will, and has decided in his own mind to keep his own virgin, does well.
7:38 So then, the one who marries(28) his own virgin does well, but the one who does not, does better.(29)
6 tn Grk "virgin," either a fiancée, a daughter, or the ward of a guardian. For discussion see the note at the end of v38.
27 tn Or referring to an engaged man: "if he is past the critical point," "if his passions are too strong." The word literally means "to be past the high point."
28 tn Or "who gives his own virgin in marriage."
29 sn 1 Cor 7:36-38. There are two common approaches to understanding the situation addressed in these verses. One view involves a father or male guardian deciding whether to give his daughter or female ward in marriage (cf. NASB, NIV margin). The evidence for this view is: (1) the phrase in v37 (Grk) "to keep his own virgin" fits this view well ("keep his own virgin [in his household]" rather than give her in marriage), but it does not fit the second view (there is little warrant for adding "her" in the way the second view translates it: "to keep her as a virgin"). (2) The verb used twice in 1Co_7:38 (γαμίζω, gamizō) normally means "to give in marriage" not "to get married." The latter is usually expressed by γαμέω (gameō), as in v36b. (3) The father deciding what is best regarding his daughter's marriage reflects the more likely cultural situation in ancient Corinth, though it does not fit modern Western customs. While Paul gives his advice in such a situation, he does not command that marriages be arranged in this way universally. If this view is taken, the translation will read as follows: "7:36 If anyone thinks he is acting inappropriately toward his unmarried daughter, if she is past the bloom of youth and it seems necessary, he should do what he wishes; he does not sin. Let them marry. 7:37 But the man who is firm in his commitment, and is under no necessity but has control over his will, and has decided in his own mind to keep his daughter unmarried, does well. 7:38 So then the one who gives his daughter in marriage does well, but the one who does not give her does better."
The other view is taken by NRSV, NIV text, NJB, REB: a single man deciding whether to marry the woman to whom he is engaged. The evidence for this view is: (1) it seems odd to use the word "virgin" (7:36; 7:37; 7:38) if "daughter" or "ward" is intended. (2) The other view requires some difficult shifting of subjects in 7:36, whereas this view manages a more consistent subject for the various verbs used. (3) The phrases in these verses are used consistently elsewhere in this chapter to describe considerations appropriate to the engaged couple themselves (cf. 7:9; 7:28; 7:39). It seems odd not to change the phrasing in speaking about a father or guardian. If this second view is taken, the translation will read as follows: "7:36 If anyone thinks he is acting inappropriately toward his fiancée, if his passions are too strong and it seems necessary, he should do what he wishes; he does not sin. Let them marry. 7:37 But the man who is firm in his commitment, and is under no necessity but has control over his will, and has decided in his own mind to keep her as his fiancée, does well. 7:38 So then, the one who marries his fiancée does well, but the one who does not marry her does better."
I'm currently convinced by the view that it is betrothed couples in view, not fathers, for the following reasons:
7:25 starts a new section concerning "virgins", and the issue is whether to marry or not. Both men and women are instructed regarding the decision they must make for themselves regarding their own marriage (v26 - 28). "Should I marry or shouldn't I" is really the only question that has been raised up to v35. (By way of response, Paul deals with more general principles and addresses people who are already married, but he remains on the same topic).
There is no linguistic clue that any new topic is introduced in v36.
The terminology remains the same from v36 onwards as it was before (apart from γαμεω/γαμιζω, which is admittedly a difficulty for this view). To use "virgin" to mean "daughter" is probably slightly strange, but to use it in this way when it has meant "betrothed" up to now is even stranger.
There is no clue in v36 at all that there has been a change in the person addressed. The main pieces of evidence appear at the end of verse 37 ("keep his own virgin") and in v38 (γαμιζω). If the section from v25 stopped at v36, it would undoubtedly be referring to the engaged men, not to fathers, and it would not seem strange. Even if it included v37, there would be very little to suggest that anyone but single men are being instructed. It's difficult to believe that from v36 Paul has started addressing a completely different group of people when there is no strong indication of that until several sentences in. The hearer would only realise that "virgin" = "daughter" once the reader got to v38, and would then have to mentally read that back into the previous few sentences — this is pretty much impossible to do, and is extremely confusing, because you suddenly don't know how far to read it back. And this switch is confusing to two groups of listeners — both single men and fathers. Fathers have been given no warning that now they need to pay attention (which is unthinkable for a letter designed to be read — any preacher will know that people tend to drift off when the subject isn't relevant to them, and need to be explicitly called back), and single men are not alerted that they are no longer being addressed (see below for more on this).
Further, we know that Paul was a preacher who would have instinctively known how to communicate, and that even in letters he was well aware of the need to make it clear very quickly if he changed the group he was addressing. See, for example, Ephesians 5 - 6 where "wives", "husbands", "children", "fathers", "slaves", "masters" are all right at the very beginning of their sentences, often without even a conjunction (asyndetic) in order to get people's attention. In all other examples in Paul's epistles that I can find he makes the audience of the commands very clear very quickly (e.g. to bishops, deacons etc).
The possibility that an engaged man who has been addressed from v25-35 might have a bad conscience about not marrying a girl he is engaged to is an obvious follow on, but switching to the father's point of view would be strange. More particularly, v36 follows directly from v35 - in v35 Paul says that he does not want to lay any restraint on the person who might want to marry, and v36 then says that he is free to do as he wills. The single men would feel sure at this point that they are still being addressed.
The idea that the decision in those times lay with the father's will seems to be contradicted by v25-35, where there is no hint of that, and in fact the opposite - it's the unmarried men/women who are being instructed about the decision they must make. If anything, Paul seems to be assuming a kind of autonomy for daughters in Christian households that may well have been uncommon in either Jewish or Greek culture.
7:36 - "having power over his own will" is a strange phrase to apply to a father, as if he was going to be giving in to intense pressure from a proposed son-in-law. On the other hand, it fits perfectly with engaged men who will often have intense desire to marry.
7:38 - if the last phrase should be translated "he that does not give her in marriage does better", then Paul leaves this idea completely unexplained — why is it better for the father to have an unmarried daughter? The emphasis up to this point has been on the individual — whether they will be able to serve the Lord better single or married, and Paul has explained that if you stay single you will do better, because you will be less torn by the complications of marriage. Paul has not been teaching that singleness is intrinsically better or more virtuous than marriage. And nowhere does he explain why keeping a daughter unmarried is a practical advantage for a father in serving God, nor is it at all obvious from the logic that has preceded. From the point of view of the daughter, the rationale for staying single (freedom to serve God) is somewhat eroded if she has equally onerous duties to her parents.
γαμεω/γαμιζω. I can't find any resources on how likely it might be that γαμιζω could mean "marry" rather than "give in marriage". Daniel B. Wallace goes for a translation in line with ESV (i.e. "marry"). In every other case of γαμιζω in the NT, it appears in a phrase with γαμεω, e.g. "marrying and giving in marriage", which makes it seem likely that this phrase doesn't limit its meaning — words can retain antiquated meanings when embedded in a phrase, meanings that are different from normal usage (e.g. in English "to leave and cleave" — people very rarely use "cleave" outside of this phrase, but when they do they are probably using the other meaning of cleave i.e. to split). Therefore it seems quite likely that in normal usage γαμιζω could have come to mean "to marry" just like γαμεω.
Further arguments regarding the sexual nature of various words in these verses are found in a Tyndale House bulletin: Puberty or Passion? The referent of υπερακμος in 1 Corinthians 7:36