Yes they did, and it was fairly easy to get divorced. Marc Antony divorced Augustus' sister in favor of Cleopatra (even though she was pregnant), and Juilus Caesar famously divorced his wife Pompeia over an incident at a religious festival, even though he had no direct evidence against her. Seutonius writes "When summoned as a witness against Publius Clodius, the paramour of his wife Pompeia, charged on the same count with sacrilege, Caesar declared that he had no evidence, although both his mother Aurelia and his sister Julia had given the same jurors a faithful account of the whole affair; and on being asked why it was then that he had put away his wife, he replied: "Because I maintain that the members of my family should be free from suspicion, as well as from accusation."" So you can take from this statement alone that there were generally few restrictions on WHY a person could divorce. The Lex Pappia (a law that aimed to strengthen and encourage marriage), while not dealing much with divorce, does state that divorced women had 6 months to re-marry or face penalty, so you can take from this that divorce was common enough to pass a penalty against a person for not remarrying. I apologize that I cannot explain the process of a divorce. I have most of my time reading Livy, Seutonius, and Polybius, all of which do not waste many (if any) words on the subject. Hopefully someone else will come along and explain what happens if one is to get divorced.