Barbier v. Connolly was a United States Supreme Court case that addressed the issue of whether a state court could issue a writ of habeas corpus to a prisoner held in federal custody. The case arose when a prisoner, Barbier, was held in federal custody in Louisiana and sought a writ of habeas corpus from the state court. The state court granted the writ, and the federal government appealed the decision to the Supreme Court. The Supreme Court held that the state court did not have the authority to issue a writ of habeas corpus to a prisoner held in federal custody. The Court reasoned that the writ of habeas corpus was a federal prerogative, and that the state court did not have the power to interfere with the federal government's authority in this matter. The Court also noted that the writ of habeas corpus was a fundamental right, and that the federal government had the exclusive power to protect this right. The Court's decision in Barbier v. Connolly established that the federal government had exclusive authority over the writ of habeas corpus, and that state courts could not interfere with this power. This decision has been cited in numerous subsequent cases, and has been used to support the idea that the federal government has exclusive authority over matters of federal law.

In the case of Barbier v. Connolly, the Supreme Court was tasked with determining whether a state court had jurisdiction to issue an injunction against certain individuals who were interfering with a public highway in Louisiana. The majority opinion held that the state court did not have such authority, as it would be unconstitutional for them to do so without congressional authorization. Justice Field dissented from this decision and argued that Congress had authorized states to protect their highways by passing legislation which allowed them to take action against those obstructing or damaging these roads. He further noted that if Congress intended otherwise they could easily pass additional laws clarifying their intent on this matter. Ultimately, he concluded that since no such law existed at the time of this case, then it should be assumed that Congress meant for states like Louisiana to have jurisdiction over protecting its highways from interference or damage caused by private parties.