An amicus brief, or amicus curiae brief, is a document containing legal arguments which is filed by a third party as a "friend of the court" to express an opinion about an issue before the tribunal. An amicus brief has the advantage of less partisanship and the freedom to embellish arguments before the court in a way that a party may not have the time or flexibility to do so. Amicus briefs filed by the United States or by a group of States may be particularly influential.
The Supreme Court has its own rules for filing an amicus brief. Most notably, an amicus brief in support of a petition for certiorari is not allowed unless notice was first given to the parties within a short period of time after the filing of the Petition. An amicus brief on the merits, after a petition for certiorari has been granted, does not require prior notice but must be filed within a short period after the side it is supporting filed its brief.
Rule 29 of the Federal Rules of Appellate Procedure govern the filing of amicus briefs before appellate courts (not the U.S. Supreme Court), with possible modification by the local rules of each court. Rule 29 itself imposes a deadline for filing an amicus brief "no later than 7 days after the principal brief of the party being supported is filed." Local rules impose stricter rules, however. In the D.C. Circuit, for example, notice of intent to file an amicus brief must be filed soon after the case is docketed on appeal, which is long before the deadline for the principal brief.
The local rules for the Court of Appeals for the Fifth Circuit are located here.
The rules in state court are different. In New Jersey, "motions for leave to appear as an amicus curiae in the Appellate Division shall be accompanied by the proposed amicus curiae brief and shall be filed on or before the day when the last brief is due from any party." N.J. Court Rules, R. 1:13-9.