What about all these last minute appeals I read about?
In criminal cases a Federal court may also review either a Federal or state court conviction after all of the usual appeals have been exhausted. This is done by a petition for "habeas corpus" -- Latin for "you have the body."
While only a tiny percentage of these petitions are granted, the reviews, especially in death penalty cases, have become highly controversial. Since an error in trying a death penalty case has extreme consequences, courts review them very scrupulously.