Topology and Combinatorics of 3 Manifolds This book is a study of combinatorial structures of 3 mani folds especially Haken 3 manifolds. Specifically it is concerned with Heegard graphs in Haken 3 manifolds i.e. with graphs whose complements have a free fundamental group. These graphs always exist. They fix not only a combinatorial stucture but also a presentation for the fundamental group of the underlying 3 manifold. The starting point of the book is the result that the intersection of Heegard graphs with incompressible surfaces or hierarchies of such surfaces is very rigid. A number of finiteness results lead up to a ri gidity theorem for Heegard graphs. The book is intended for graduate students and researchers in low dimensional topolo gy as well as combinatorial theory. It is self contained and requires only a basic knowledge of the theory of 3 manifolds