The appellate jurisdiction of the Supreme Court refers to the authority of the Court to review the decisions of lower courts. Most of the cases that the Supreme Court hears are appeals from lower courts. The Constitution states that the Supreme Court has both original and appellate jurisdiction. Original jurisdiction means that the Supreme Court is the first and only court to hear a case, and it is limited to disputes between states or disputes arising among ambassadors and other high-ranking ministers. On the other hand, appellate jurisdiction means that the Court has the authority to review the decisions of lower courts.

In the United States, Congress has authorized Supreme Court review of decisions of state courts and lower federal courts through two procedural mechanisms: appeals and petitions for a writ of certiorari. The Court has discretion to grant or deny review via a petition for a writ of certiorari, while it is required to exercise jurisdiction over cases properly before it on direct appeal. Over time, Congress has limited the types of cases subject to direct appeal to the Supreme Court, rendering more cases subject to discretionary review via certiorari.

In Texas, the Supreme Court has statewide, final appellate jurisdiction in civil and juvenile cases, and original jurisdiction to issue writs. It has general responsibility for the efficient operation of the Texas judicial system and is empowered to make and enforce all necessary rules of civil trial practice and procedure, evidence, and appellate procedure.

In summary, the appellate jurisdiction of the Supreme Court allows it to review the decisions of lower courts, and most of the cases that the Court hears are appeals from lower courts.