What does the Bible say about at what age to get married?

As far as I can find in the Bible it seems that a boy and girl can marry at age 12 and 13 years old, due to their physical developments (ie.puberty). Is that acceptable in today's society?

Simply put, the Bible provides no specific direction regarding the proper age for a man or woman to enter into marriage. This means the believer has liberty in this area: we may marry at whatever age seems best to us. On the other hand, the believer is also required to obey the governing authorities over us:

Rom. 13:1 Every person is to be in subjection to the governing authorities. For there is no authority except from God, and those which exist are established by God.
Rom. 13:2 Therefore whoever resists authority has opposed the ordinance of God; and they who have opposed will receive condemnation upon themselves.
Rom. 13:3 For rulers are not a cause of fear for good behavior, but for evil. Do you want to have no fear of authority? Do what is good and you will have praise from the same;
Rom. 13:4 for it is a minister of God to you for good. But if you do what is evil, be afraid; for it does not bear the sword for nothing; for it is a minister of God, an avenger who brings wrath on the one who practices evil.
Rom. 13:5 Therefore it is necessary to be in subjection, not only because of wrath, but also for conscience’ sake.
Rom. 13:6 For because of this you also pay taxes, for rulers are servants of God, devoting themselves to this very thing.
Rom. 13:7 Render to all what is due them: tax to whom tax is due; custom to whom custom; fear to whom fear; honor to whom honor.

According to scripture, every Christian must obey the laws of the place where they live, unless a law directly contradicts the commandments of scripture. Therefore, we must obey the laws concerning marriage, and every civilized community has a minimum marrying age. Christians must abide by that law or else sin in violation of Romans 13. If a person decides to violate the law in this matter or any other, Paul says we should expect the governing authorities to punish us accordingly, which is God’s will for those who rebel in this way.