In addition to theoretical material, this book contains many helpful exercises, suggestions for further reading and a reasonable bibliography. A reader may choose to concentrate on topological groups, locally compact groups or topological fields, rings or semigroups (but a continuous reading may be more convenient from several points of view). After some standard basic material (topology, groups, topological transformation groups), the text continues with the Haar integral and its application to linear representations. A chapter on categories contains representations of compact groups as projective limits of Lie groups (or of finite powers of circles for Abelian groups). The section on locally compact groups deals with Pontryagin duality, approximation of locally compact Abelian groups by Lie groups and a discussion of maximal compact and vector subgroups. It continues with results on automorphism groups of locally compact Abelian groups, results on locally compact rings and fields, and homogeneous topological groups. The short section on locally compact semigroups contains some basic material. The last chapter deals with Hilbert's fifth problem. It contains results on approximation of locally compact groups, their dimension, simplicity, metrizability, connectedness, etc.