One and a half centuries of invariant theory

Invariant Theory is a branch of algebra that emerged about 150 years ago as a study of polynomials that are transformed in a prescribed way under nondegerate linear transformations of variables.This theory went through several periods of rises and falls and nowadays it is flourishing again mainly because of deep, mutually fruitful connections with a number of disciplines (algebraic groups, Lie groups, algebraic geometry, representation theory, commutative algebra, homologocal algebra, Galois theory, ring theory, combinatorics, coding theory) and famous mathematical problems (Hilbert's 14th and 13th problems). In fact, Invariant Theory gave birth to some of them (commutative algebra and homological algebra). The talk is intended to describe the main streams and results of this theory, from the beginning to the present time.