Invariant Random Fields on Spaces with a Group Action 1.Introduction 2.Spectral Expansions 3.L2 Theory of Invariant Random Fields 4.Sample Path Properties of Gaussian Invariant Random Fields 5.Applications A.Mathematical Background References Index. The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics including probability theory differential geometry harmonic analysis and special functions. The present volume unifies many results scattered throughout the mathematical physical and engineering literature as well as it introduces new results from this area first proved by the author. The book also presents many practical applications in particular in such highly interesting areas as approximation theory cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes statistics functional analysis astronomy and engineering. .