Ring theorists and researchers in invariant theory and operator algebra met at Bowdoin for the 1984 AMS-IMS-SIAM Joint Summer Research Conference to exchange ideas about group actions on rings and to stimulate general interaction among their various fields. Collecting together the proceedings of that meeting, this interdisciplinary volume reveals many topics common to the three fields: K-theory, dual actions, semi-invariants, and crossed products. A high percentage of the papers here are expository; of particular significance are the papers by the four survey lecturers, Edward Formanek on noncommutative invariant theory; Melvin Hochster on invariant theory of commutative rings; D. S. Passman on algebraic crossed products; and Marc A. Reiffel on $K$-theory of crossed products of C*-algebras by discrete groups.