The Bible is very clear about the fact that God abhors divorce. However, this does not mean that there are not some valid reasons that a Biblical divorce is allowed. Divorce was present in the Old Testament under the law, and Jesus spoke about divorce in the New Testament also.
Matthew 5:32 says that, "Any man who divorces his wife except for sexual immorality causes his wife to commit adultery." Sexual immortality includes fornication, prostitution and adultery. From this Greek translation, many presume that Jesus was saying that a Biblical divorce is allowable if there is some type of sexual sin by one of the marriage partners. However, the translation does focus on the word allowable which means it is okay but not a requirement. Even in the old testament under the law, there were cases where God asked a man to stay with his unfaithful wife. One example of this is in the book of Hosea where Hosea is instructed to take back his wife, who is a prostitute.
First Corinthians 7:15 says, "but if the unbeliever leaves, let him do so. A believing man or woman is not bound in such circumstances; God has called us to live in peace." This verse can be looked at in two ways. The first is that the abandonment has to be from an unbeliever in order for it to be Biblical. The second way to look at this is that abandonment under any circumstances is an allowable reason for divorce.
It is hard to believe that the Bible would not include abuse of the spouse or the children as a reason for divorce. This is not specifically mentioned in the Bible. However, common sense would advocate that, in any case of abuse, safety needs to come first and at the very least separation from that spouse is necessary. Divorce may not be sanctified in this case, but safety of the abused spouse and children must take precedence.
Remarriage after a divorce is a cloudy subject. Because the Bible is specific about the reasons for divorce, some argue that the reasons for remarriage are specific. Also It should be pointed out that just because the Bible says divorce is allowable, it does not mean that the Bible requires it. Most churches and Christian counselors will advise couples to try and work out all problems as they teach forgiveness and reconciliation through grace.