The Supreme Court has original and exclusive jurisdiction over cases between two or more states. it can decline some of those cases.
In 1906, the Court started to attend cases of individuals for contempt of court in the United States. In all other cases, the Court has only appellate jurisdiction, including the ability to issue writs of mandamus and writs of prohibition to lower courts.
Also, the Supreme court can appellate to another court, from the country or from another country with an issue related. The Court recognizes some circumstances where it is appropriate to hear a case that is seemingly moot.