In the U.S., each U.S. State has a State Supreme Court, and for the United States as a whole, the highest Court is the Supreme Court of the United States. The Supreme Court of the United States can, by the U.S. Constitution, review any case or controversy, typically as to Constitutional issues, to settle uncertainties as to the law and further define the legal meaning of the U.S. Constitution.
However, the Court does usually defer to State Supreme Court decisions having adequate and independent State grounds. A State Supreme Court has to explicitly specify in a ruling it hands down that the ruling is based on adequate and independent State grounds, according to what in law constitutes adequate and independent State grounds. For instance, a State Constitutional ruling interpreting eminent domain law can be broader than the support set forth in Amendment V of the U.S. Constitution, but not narrower.
The State Supreme Courts don't just get to say "our decision is final and cannot be appealed for writ of certiorari to the Supreme Court of the United States because we say so." That is, adequate and independent State grounds do not exist where inconsistent with the U.S. Constitution and the Court's interpretation of it. Where State Constitutions mirror the U.S. Constitution, the State Constitution provisions cannot be interpreted more narrowly than how the U.S. Constitution provisions are interpreted, and then "justified" as having been done on "adequate and independent State grounds". Other State Constitutional provisions that are unconstitutional by the U.S. Constitution also will be struck down when brought to the Court as a constitutional "case or controversy". This derives from the theory of dual sovereigns in American federalism.
The Court's authority for this derives from U.S. Const., Art VI, Cl. 2:
"This Constitution, and the Laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the constitution or laws of any state to the contrary notwithstanding."