If the question is asking about appeals of state supreme court rulings, then the first federal level of appeal would be at the U.S. Court of Appeals for whatever Federal Judicial Circuit the state was located in.
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Appeals from the Courts of Military Review are heard by the United States Court of Appeals for the Armed Forces (CAAF). This court is specifically tasked with reviewing decisions made by the Courts of Military Review, which handle cases involving military personnel. The CAAF's rulings can be further appealed to the Supreme Court of the United States, but only if the Supreme Court chooses to hear the case.
The US Courts of Appeals Circuit Courts hear appeals from US District Courts under their appellate jurisdiction. They do not hear cases of original jurisdiction (trials).Appellate courts review the written account of the case with the appeal lawyer's statements of the mistakes made during the trial or of misapplied laws or constitutional challenges.Example: At a simpler level, I used to explain to my students that the judge supervises the lawyers and the appeals court supervises the judge. The judge decides when lawyers argue with each other and keeps the trial flowing. Thus if the judge sides with one lawyer the appeals court can say he made a mistake. The judge stops a particular line of questioning. The appeals court can say he should not have done so. If the judge gives incorrect incorrect instructions to the jury, the appeals court can say he did it wrong.
Decisions made by a state appeals court are binding on lower courts within the same state, meaning that trial courts must follow the legal principles established by the appeals court. Additionally, the decisions are typically binding on subsequent cases involving similar facts and legal issues within that jurisdiction. However, these decisions do not bind other state appeals courts or federal courts, as they operate under different jurisdictions and legal precedents.
US Special Courts or courts of special jurisdiction hear a variety of cases, each related to a particular subject matter. For example, US Bankruptcy Courts hear bankruptcy cases; US Tax Court hears federal tax cases; the US Court of Federal Claims hears cases involving monetary damages against the US government; the US Court of Appeals for Veteran's Claims hears appeals of decisions made by the Board of Veteran's Appeals, and so on. Military courts are also considered special courts.
There are several courts to which an appeal may be made, depending on the circumstances. The following site contains a discussion:
Under most circumstances the US District Courts are the federal trial courts of general jurisdiction; however, they also hear appeals of federal agency decisions made by Administrative Law Judges (for example, appeals of Social Security Disability cases).
The Supreme Court and other appeals courts usually do not hear any cases that involve issues of fact; instead, they focus on questions of law. These courts primarily review decisions made by lower courts to determine if legal principles were correctly applied. Additionally, they typically avoid cases that are deemed moot or lack a substantial federal question.
Becasue they are the lowest level state courts in the state judicial system - courts of "original jurisdiction' which hear cases within their assigned judicial "circuit."
Trial courts typically do not have appellate jurisdiction; their primary function is to hear cases for the first time and make determinations of fact and law. Instead, appellate jurisdiction is held by higher courts that review the decisions made by trial courts. Examples of courts with appellate jurisdiction include state appellate courts and federal courts of appeals. In contrast, trial courts, such as district or circuit courts, focus on original jurisdiction.
Twelve of the thirteen US Courts of Appeal Circuit Courts have territorial jurisdiction over appeals of general criminal and civil cases original tried in US District Courts. The US Court of Appeals for the Federal Circuit has nationwide subject matter jurisdiction over cases originally heard in special courts, such as the US Court of Federal Claims, the US Court of International Trade, the Court of Veterans' Claims, and cases involving patent disputes, decisions made by government boards and departments, and other entities.
Many US jurisdictions title their appellate court a Court of Appeal or Court of Appeals. Historically, others have titled their appellate court a Court of Errors (or Court of Errors and Appeals), on the premise that it was intended to correct errors made by lower courts. Examples of such courts include the New Jersey Court of Errors and Appeals (which existed from 1844 to 1947), the Connecticut Supreme Court of Errors (which has been renamed the Connecticut Supreme Court), the Kentucky Court of Errors (since renamed the Kentucky Supreme Court), and the Mississippi High Court of Errors and Appeals (since renamed the Supreme Court of Mississippi). In some jurisdictions, courts able to hear appeals are known as an Appellate Division. Depending on the system, certain courts may serve as both trial courts and appellate courts, hearing appeals of decisions made by courts with more limited jurisdiction. Some jurisdictions have specialized appellate courts, such as the Texas Court of Criminal Appeals, which only hears appeals raised in criminal cases, and the United States Court of Appeals for the Federal Circuit, which has general jurisdiction but derives most of its caseload from patent cases, on the one hand, and appeals from the Court of Federal Claims on the other.