This textbook for invariant theory and research monograph that introduces a new approach to the algorithmic side of invariant theory. The Groebner-based method is the main tool by which the central problems in invariant theory become amenable to algorithmic solutions. Students should find the book useful as an introduction to this classical and new area of mathematics. Researchers in mathematics, symbolic computation and computer science should get access to a wealth of research ideas, hints for applications, outlines and details of algorithms, worked-out examples and research problems.