The invariant theory of non-reductive groups has its roots in the 19th
century but has seen some very interesting developments in the past
twenty years. This book is an exposition of several related topics
including observable subgroups, induced modules, maximal unipotent
subgroups of reductive groups and the method of U-invariants, and the
complexity of an action. Much of this material has not appeared
previously in book form. The exposition assumes a basic knowledge of
algebraic groups and then develops each topic systematically with
applications to invariant theory. Exercises are included as well as many
examples, some of which are related to geometry and physics.