  Unipotent flows are well-behaved dynamical systems. In particular, Marina
Ratner has shown that the closure of every orbit for such a flow is of a nice
algebraic (or geometric) form. After presenting some consequences of this
important theorem, these lectures explain the main ideas of the proof. Some
algebraic technicalities will be pushed to the background.
  Chapter 1 is the main part of the book. It is intended for a fairly general
audience, and provides an elementary introduction to the subject, by presenting
examples that illustrate the theorem, some of its applications, and the main
ideas involved in the proof. It should be largely accessible to second-year
graduate students.
  Chapter 2 gives an elementary introduction to the theory of entropy. Chapter
3 presents some basic facts of ergodic theory, and Chapter 4 lists some facts
about algebraic groups.
  Chapter 5 presents a fairly complete (but not entirely rigorous) proof of the
measure-theoretic version of Ratner's Theorem. (We follow the approach of
G.A.Margulis and G.Tomanov.) Unlike the other chapters, it is rather technical.
