It is the continuation of the deep branch of the radial nerve, after this has crossed the supinator muscle.

In other words, it supplies the following muscles: The posterior interosseous nerve provides proprioception to the joint capsule of the distal radioulnar articulation, but not pain sensation.

[1] The posterior interosseous nerve may be entrapped at the arcade of Frohse, which is part of the supinator muscle.

This nerve can be injured in Monteggia fracture due to dislocation of the proximal head of radius bone.

This article incorporates text in the public domain from page 944 of the 20th edition of Gray's Anatomy (1918).