This volume contains the expository lectures and a selection of short communications presented at the Summer School Algebraic and Geometric Methods in Mathematical Physics, held in Kaciveli, Crimea, Ukraine, in September 1993. The contributions, by leading experts in the various fields, review the state of the art in many important branches of modern mathematical physics. Special emphasis is given to certain aspects of quantum groups and conformal field theory, spectral theory of differential and pseudodifferential operators, nonlinear integrable PDEs and related problems of algebra, geometry and analysis. A number of topics of current interest is also discussed, such as nonlinear problems of mathematical economics, direct and inverse problems of spectral theory, mathematical statistical mechanics, etc. Audience: Researchers and graduate students in group representations, spectral theory, nonlinear equations, integrable systems, mathematical quantum field theory and statistical mechanics.