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Abstract. The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure theoretic techniques. These notes provide an introduction to some of the important methods, major developments, and open problems in the subject. They are slightly expanded from lectures of R.J.Zimmer at a CBMS Conference at the University of Minnesota, Minneapolis, in June, 1998. The main text presents a perspective on the field as it was at that time, and comments after the notes of each lecture provide suggestions for further reading, including references to recent developments, but the content of these notes is by no means exhaustive.

Table of Contents

Lecture 1. Introduction

Lecture 2. Actions in Dimension 1 or 2

Lecture 3. Geometric Structures

Lecture 4. Fundamental Groups I

Lecture 5. Gromov Representation

Lecture 6. Superrigidity and First Applications

Lecture 7. Fundamental Groups II (Arithmetic Theory)

Lecture 8. Locally Homogeneous Spaces

Lecture 9. Stationary Measures and Projective Quotients

Lecture 10. Orbit Equivalence

Appendix: Background Material