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Divorce is the permanent dissolution of a marriage. Divorce can only be obtained through a court order. There is no such thing as "common-law divorce," so if you are in a common-law marriage, and want to end the relationship, you will have to file for divorce.
Modernly, every state but one (New York) allows for divorce on a "no fault" basis. This means that virtually anything can serve as grounds for divorce. If the relationship has simply run its course, the parties can cite irreconcilable differences. While the other traditional grounds for divorce are still recognized, they're rarely necessary, since "no fault" divorce covers every possible situation.
In New York, however, you will still need to allege, and prove, that at least one of the traditional grounds for divorce has taken place. Grounds for divorce include cruelty, confinement in prison, inability to consummate the marriage, abandonment, and infidelity.
Some states have done away with these traditional grounds for divorce altogether, completely replacing them with the no-fault system, reasoning that they're redundant, since one no longer needs to cite a reason for getting a divorce. However, the majority of states still maintain the traditional "fault" system alongside their no-fault system.
Division of property at divorce
When two people are married for even a moderate amount of time, they are likely to accumulate a good deal of money and personal property together. If a marriage is going smoothly, they usually don't think about which spouse individual pieces of property belong to: it's just "theirs."
However, in the event of a divorce, this can become a very contentious issue. In the U.S., there are two basic systems of marital property ownership: separate property, and community property.
In a separate property state, any money that one spouse earns at a job or in business, or otherwise acquires, belongs solely to that spouse. The same goes for any property bought with that money.
If one spouse is the primary breadwinner, and the other is a homemaker, this will lead to a lopsided arrangement, in which one spouse owns virtually all the property that they both use.
On the other hand, if both spouses work, and deposit their money into a joint account, determining exactly who owns what can be very difficult (but quite lucrative if you happen to be a forensic accountant).
In the event of a divorce, property is subject to "equitable distribution" which means that it will be divided among the spouses in a way that the court deems to be fair. Note that "equitable" does not mean "equal" – it simply means fair, and those aren't always the same thing.
A court will almost always give some of one spouse's property to the other, and in determining how much to give, it will look at many factors, such as whether or not the spouse gave up a lucrative career to tend the home, what type of lifestyle that spouse has become accustomed to, etc.
In a community property system, any money or property acquired by either spouse during marriage is owned equally by both of them – each spouse owns an undivided 50% interest. There are exceptions, however: property which is a gift to one of the spouses, or which is acquired by inheritance or devise, is owned solely by that spouse.
All property acquired by either spouse before the marriage is also owned individually, and is not community property.
At divorce, all community property, by default, is split 50/50. With money, this is easy to do. With things that can't be "split" like houses and personal property, the property is typically sold, with the spouses splitting the proceeds. Alternatively, the spouses could agree on who gets to keep what ("you keep the house, I keep the cars" etc.). Community property is just the default arrangement in the states that have it. If the spouses come to an agreement amongst themselves about how property is to be split, even if it doesn't result in a 50/50 division of marital property, a court will not interfere.
Furthermore, a couple can agree to convert a piece of community property into separate property of either spouse. They can also do the same with separate property, converting it to community property by agreement.