A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving nonlinear integrable equations for various physical systems. Physicists and engineers involved in studying solitons, phase transitions, or dynamical (gyroscopic) systems, and mathematicians with some background in algebraic geometry and abelian and automorphic functions, are the targeted audience. This book is suitable for use as a supplementary text to a course in mathematical physics.