This volume presents the proceedings of the workshop 'Harmonic Functions on Graphs' held at the Graduate Center of CUNY in the fall of 1995. The main papers present material from four minicourses given by leading experts: D. Cartwright, A. Figa-Talamanca, S. Sawyer and T. Steger. These minicourses are introductions which gradually progress to deeper and less known branches of the subject. One of the topics treated is buildings, which are discrete analogues of symmetric spaces of arbitrary rank; buildings of rank are trees. Harmonic analysis on buildings is a fairly new and important field of research.One of the minicourses discusses buildings from the combinatorial perspective and another examines them from the $p$-adic perspective. The third minicourse deals with the connections of trees with $p$-adic analysis. And the fourth deals with random walks, i.e., with the probabilistic side of harmonic functions on trees. The book also contains the extended abstracts of 19 of the 20 lectures given by the participants on their recent results. These abstracts, well detailed and clearly understandable, give a good cross-section of the present state of research in the field.