There is no automatic right to review by the Supreme Court.  A party that desires to have their case heard by the Court must file what is called a Petition for a Writ of Certioari.  The Justices decide based on that petition, as well as any brief in opposition thereto filed by an opposing party, whether they will hear the case.  If the Justices grant the Petition for a Writ of Certioari, the parties are directed to file briefs and oral argument may be set.

There is a very strict time limit of ninety (90) days in which a party must file the Petition.  The ninety days start running from the time of the entry of final judgment in the lower court.