This paper is an expanded version of remarks delivered by the authors in lectures at the June, 1990 Amherst conference on Quantum Groups. There we were asked to describe, in so far
as possible, the basic principles and results, as well as the present state, of algebraic deformation theory. So this paper contains a mixture of the old and the new. We have attempted to provide a fresh perspective even on the more "ancient" topics, highlighting problems and conjectures of general interest throughout. We trace a path from the seminal case of associative algebras to the quantum groups which are now driving deformation theory in new directions. Indeed, one of the
delights of the subject is that the study of btalgebra deformations has led to fresh insights in the classical case of associative algebra — even polynomial algebra! — deformations.