There are some matters in the exclusive original jurisdiction of the U.S. Supreme Court, which consists of suits between U.S. states and/or foreign states with each other and suits involving diplomats. On average, one or two such suits are filed each year.
The original jurisdiction in these cases is created by Article III of the United States Constitution (Section 2, Clause 2), and is exclusive in these cases by virtue of the 1789 Judiciary Act. The 11th Amendment also plays a role in this analysis.
The original jurisdiction of the U.S. Supreme Court is set forth in Article III, Section 2, Clause 2 of the United State Constitution, which states:
In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court
shall have original jurisdiction. In all the other cases before
mentioned, the Supreme Court shall have appellate jurisdiction, both
as to law and fact, with such exceptions, and under such regulations
as the Congress shall make.
Basically, these are cases of a state against the United States and/or another state, often regarding boundary disputes, interpretations of interstate compacts, or water rights, or cases involving diplomats adjudicating the extent of diplomatic immunity.
(Incidentally, almost all, but not all of the cases in the U.S. Supreme Court's appellate jurisdiction are discretionary, but appeals to the U.S. Supreme Court from three judge panels ruling in certain election law cases are of right.)
How rare are these suit in the original jurisdiction of the U.S. Supreme Court?
On average, the U.S. Supreme Court has considered about one per year on the merits since it came into existence. This has been between 0.5% and 2% of the U.S. Supreme Court's overall caseload of merits cases in modern times. (These days the U.S. Supreme Court typically decides about 70 case a year on the merits and evaluates about 5,000 certiorari petitions each year.)
Between 1789 and 1959, the Court issued written opinions in only 123
original cases. Since 1960, the Court has received fewer than 140
motions for leave to file original cases, nearly half of which were
denied a hearing. The majority of cases filed have been in disputes
between two or more states. The Court has generally accepted state
party cases dealing with boundary and water disputes, but it has been
much less likely to field original cases dealing with contract
disputes and other subjects not deemed sufficiently substantial for
the Court's resources.
In practice, when a case in the U.S. Supreme Court's original jurisdiction is filed, it is almost always assigned immediately to a "special master" who develops the case until it is ready for U.S. Supreme Court review.
The U.S. Supreme Court briefly allowed non-residents of a state to sue state governments in its original jurisdiction, but this authority was quickly eliminated by the passage of the 11th Amendment (which has also been interpreted as codifying the principal of a state's sovereign immunity from suits by its own citizens outside its own courts without its consent).
In the Judiciary Act of 1789, Congress made the Supreme Court's
original jurisdiction exclusive in suits between two or more states,
between a state and a foreign government, and in suits against
ambassadors and other public ministers. The Supreme Court's
jurisdiction over the remainder of suits to which a state was a party
was to be concurrent, presumably with state courts since the statute
did not expressly confer these cases upon the inferior federal courts.
Notably, this exclusivity rule does not apply to suits between a state and the United States, or to suits brought by states against non-residents (whether or not they are U.S. persons), although such suits have been interpreted to be within the original jurisdiction of the U.S. Supreme Court:
In the 1892 case of United States v. Texas , Justice John Marshall
Harlan ruled that since the federal judicial power extended to "cases
in which the United States was a party," and the Court was granted
jurisdiction over cases to which a state was a party, the Court would
take jurisdiction in a United States suit against a state. Such suits
by the United States increased after the 1890s and usually involved
disputes with states over land, though in the late twentieth century
they also included a few suits to enforce provisions of the Federal
Voting Rights Act.
As a result of the 11th Amendment and the 1789 Judiciary Act, the U.S. District Courts only have jurisdiction over states when they consent to suit in the forum (usually for federal law cases involving bankruptcies), or when the United States is a party and no private individuals are parties (as in the federal district court case Arizona v. United States mentioned in the comments) since U.S. District Courts have jurisdiction over all suits to which the United States is a party.
U.S. District Court jurisdiction is limited, however, by doctrines of state sovereign immunity, to cases seeking only injunctive relief that seek to enforce the 14th Amendment to the United States Constitution (although this is circumvented, in part, by bringing suits against state officials as opposed to state governments themselves). So basically, states themselves can only be sued in U.S. District Courts by the United States for injunctive relief under the 14th Amendment. (For this purpose, unlike many other purposes under the U.S. Constitution, the term "state" does not include local governments which can be sued in federal court, and often are sued there for civil rights violations.)
Also, despite the seemingly mandatory nature of the U.S. Supreme Court's original jurisdiction, it declines to hear about half of the cases presented to it in that capacity.
The Supreme Court further limited its original docket by declaring
that it would exercise discretion over whether to hear cases even if
they were legitimately within the Court's jurisdiction. In a series of
cases in 1971, including Ohio v. Wyandotte Chemicals Corp ., the Court
declined to hear environmental pollution claims brought by states
against corporations that dealt with complex and technical factual
questions. The justices ruled that the states had other available
forums to bring their claims and that the cases were not "appropriate"
for the Court in light of its primary function as the nation's highest
appellate tribunal. The Court resolved to examine the "seriousness and
dignity" of claims so as to preserve its resources for consideration
of appeals involving federal questions. The Supreme Court soon
expanded its appropriateness doctrine to decline to hear some cases
between two states, even where the Court's jurisdiction was exclusive.
The U.S. Supreme Court also has the statutory authority (almost never used) under the All Writs Act (28 U.S.C. § 1651) to issue writs of habeas corpus filed directly in the U.S. Supreme Court rather than a lower court, although strictly speaking it doesn't add to the U.S. Supreme Court's jurisdiction. About 60 habeas corpus cases are filed directly in the U.S. Supreme Court in its original jurisdiction each year, although very few are granted.
The text of the All Writs Act is as follows:
(a) The Supreme Court and all courts established by Act of Congress
may issue all writs necessary or appropriate in aid of their
respective jurisdictions and agreeable to the usages and principles of
(b) An alternative writ or rule nisi may be issued by a justice or
judge of a court which has jurisdiction.
According to Wikipedia (with appropriate citation to authority):
Application of the All Writs Act requires the fulfillment of four
The absence of alternative remedies—the act is only applicable when
other judicial tools are not available.
An independent basis for jurisdiction—the act authorizes writs in aid
of jurisdiction, but does not in itself create any federal
Necessary or appropriate in aid of jurisdiction—the writ must be
necessary or appropriate to the particular case.
Usages and principles of law—the statute requires courts to issue
writs "agreeable to the usages and principles of law".