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