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The Supreme Courts of New York are the trial courts, the lowest level courts of general jurisdiction in the New York State judiciary.
New York State designated their 62 trial courts (courts of original, general jurisdiction) "The Supreme Court of the State of New York." The State's highest appellate court is the Court of Appeals.
The US Supreme Court is not located in New York. New York does have a trial court system which is called the Supreme Court, even though it is not truly supreme in the sense that the US Supreme Court is. It is not an appellate court like the US supreme Court is. They named it that because in early days, New York had a fractured system of trial courts, some with limited subject matter jurisdiction, some with limited regional jurisdiction. The New York Supreme Court became the unified statewide trial court, in effect supreme over all the little courts. It is not a good choice of names.
They have original jurisdiction over the cases emanating from the lower courts within their assigned circuit.
Although some New York jurisdictions may have additional trial courts, the three main trial courts in the New York state court system are Supreme Courts, Family Courts, and Surrogate's Courts. These three types of courts are located in every county. For more information on New York courts, visit the New York Courts Directory related link.
New York State Circuit Courts ended in 1847.
New York State Circuit Courts was created in 1823.
US admiralty courts, also known as federal maritime courts, have jurisdiction over cases involving maritime law, including disputes related to ships, seamen, cargo, and marine insurance. These courts are found in key coastal cities such as New York, New Orleans, and San Francisco.
This question can't be answered without knowing the state in which these courts reside. States have different names for courts that perform the same function. For instance, the court of general jurisdiction, where serious criminal trials are heard, is the supreme court in New York, but called the superior court in California.
They are in different places on the hierarchy of jurisdiction. Appellate jurisdiction is higher. Courts with appellate jurisdiction can hear appeals, whereas courts with original jurisdiction can hear cases for the first time.
A supreme court is an appellate court existing in most of the states. In the federal court system, and in most states, it is the highest appellate court or court of last resort. In others (such as New York) the supreme court is a court of general original jurisdiction, possessing also (in New York) some appellate jurisdiction, but not the court of last resort.
It really depends what kind of courts you are talking about and where the courts are. If the courts are federal, then a district court is a trial court and a circuit court is an appeals court, which may review a trial decision from a district court. For state courts, the difference between a district court and a circuit court will depend on what state the courts are in. Many states have courts called "district court" and "circuit court," but what kinds of cases these courts handle differs state to state.