There seems to be a great deal of confusion about Catholic marriages on this thread.
Neither adultery nor fornication are bars to marriage in the Catholic church. Those presenting themselves to be married are expected to go to confession prior to the wedding ceremony and confess their sins, which most definitely do include adultery and fornication, repent, be absolved and do penance. The priest who is to officiate at the wedding is within his rights to ask whether they have confessed etc. and to refuse to marry them if they have not. They do not need to have used him as their confessor, though.
You are all correct regarding divorce as a bar to marriage.
According to Canon Law no-one may marry who has a living spouse and a divorced husband or wife counts as a spouse.
The divorced party to a new marriage can ask a bishop's court to look into the grounds for the former marriage and judge whether it was or was not valid in Canon Law.
This process takes time depending on how many cases the court has to deal with. These days most priests preparing such a case state the grounds for invalidity being the presumption that the parties married without a proper awareness of the fact that a Catholic marriage is a sacrament and with no intention of abiding by the condition of the service that states that marriage lasts "until death do us part"
If for this, or any other of the reasons that invalidate a marriage in Canon Law, the person's former marriage or marriages are deemed invalid, he or she can then marry in the Catholic church.
If, which is usually the case, such a marriage was valid by the secular law of the country any children of it are legitimate and their status is not changed because Canon law deems their parent's marriage invalid.
A marriage can only be annulled if it was never consummated, which necessitates one at least of the parties to it proving, with the help of medical evidence that he or she was not physically capable of consummating the marriage, or had no opportunity to do so.
Annulments are relatively rare these days, where very few women of any age can proof themselves intact virgins, which is after all the easiest way of proving non-consummation.
If neither party to the intended marriage has been married before, then if one of them is a Catholic they can and should marry in a Catholic ceremony, as Catholics may not marry outwith the Catholic church.
If you do so, you are deemed to have excommunicated yourself.
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