Marriage and divorce are not recognized if there are no witnesses. What else cannot be done without witnesses? Why are witnesses important?
This is a very open-ended question. All crimes need to be testified to either by the criminal themselves or by others. Certain sins like zina require four witnesses who witnessed the actual act taking place. Others like divorce and business deals require two. There are some where the testimony of women is inadmissible and others where two women replace one man, and yet others where a woman's testimony overrides that of a man. Witnesses would be required in muddied up cases to prove virginity, child birth, inheritance, murder, apostasy.
Witnesses are very important in a Deen that is from Allah, the Just, to prevent injustices from taking place and for Truth to prevail.