Courts that oversee bankruptcy filings are part of the federal court system, and are generally referred to as U.S. Bankruptcy Courts.
In New York, as in all states, courts that oversee bankruptcy filings are indeed part of the federal court system, specifically designated as U.S. Bankruptcy Courts. These courts handle all matters related to bankruptcy filings, including petitions and proceedings under the various chapters of the Bankruptcy Code. The U.S. Bankruptcy Courts operate under the umbrella of the U.S. District Courts, which are the general trial courts of the federal court system. New York is divided into four bankruptcy districts: the Eastern, Southern, Northern, and Western Districts of New York, each with its own bankruptcy court(s). The federal nature of bankruptcy law means that while the process is consistent across the United States, local rules and procedures can vary from district to district within the state.