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The Supreme Court is the highest appellate court in the United States. It does not try cases, in the standard sense of the word, but only reviews lower court decisions. Usually, cases heard by the Supreme Court involve issues of Constitutional law or federal legislation.
The Supreme Court hears any cases that involve the interpretation of the Constitution.
The Supreme Court decides cases that are appealed by a lower court; a lower court has made a decision and one of the parties feels strongly enough that the decision was wrong that they make an appeal to the Supreme Court. The Supreme Court reviews the cases and determines which ones they will hear, they have the ability to decline to review a case. The Supreme Court doesn't hear only appeals, there are situations where it is the court of original jurisdiction. In situations where there is a disagreement between states, the Supreme Court has the authority to decide.
Supreme Court of the United States
There are two special cases that start trial in the United States Supreme Court. Cases involving foreign officials and cases in which a state is a party originate in the Supreme Court.