The New York State Divorce Court handles divorce cases in New York, aiming to provide a fair and efficient process. It is part of the New York State Unified Court System, which includes the Supreme Court, Family Court, and Appellate Courts. The court follows a specific procedure, including filing a divorce petition, responding to it, discovery, mediation, negotiation, and trial. To file for divorce, individuals must meet residency, grounds for divorce, and financial disclosure requirements.