What kinds of court cases can be heard in either state or federal courts?

State courts can hear any almost any kind of case without limitation. They are the only courts of general jurisdiction. Federal Courts are courts of limited jurisdiction. Incidentally, ';jurisdiction'; in this answer can be equated to ';power.'; There are only a few cases that have exclusive Federal Court jurisdiction. The most common are Indian law issues and offenses occurring on a military base or federal land. Even DUI's on a military base go to federal court. Many people mistakenly believe the U.S. Constitutional cases MUST go to federal court. That is incorrect. These cases can be removed to federal court, but state courts have concurrent jurisdiction.