J. Kung and G.-C. Rota, in their 1984 paper, write: "Like the Arabian
phoenix rising out of its ashes, the theory of invariants, pronounced
dead at the turn of the century, is once again at the forefront of
mathematics". The book of Sturmfels is both an easy-to-read textbook for
invariant theory and a challenging research monograph that introduces a
new approach to the algorithmic side of invariant theory. The Groebner
bases method is the main tool by which the central problems in invariant
theory become amenable to algorithmic solutions. Students will find the
book an easy introduction to this "classical and new" area of
mathematics. Researchers in mathematics, symbolic computation, and
computer science will get access to a wealth of research ideas, hints
for applications, outlines and details of algorithms, worked out
examples, and research problems.