This advanced graduate level monograph is the first to present a wide variety of critical point theorems in a unified framework. The author not only employs Morse theory as a tool to study multiple solutions to differential equations arising in the calculus of variations, but covers a broad range of applications to semilinear elliptic PDE, to dynamical systems and symplectic geometry, and to geometry of harmonic maps and minimal surfaces. Critical groups for isolated critical points or orbits - which provide more information than the Leray-Schauder index - are introduced. Topics covered include basic Morse theory and its various extensions, minimax principles in Morse theory, and applications of semilinear boundary value problems, periodic solutions of Hamiltonian systems, and harmonic maps. In a self-contained appendix, the author presents Witten's proof of Morse inequalities. Containing several new results, this volume will be attractive and germaine to researchers and graduate students working in nonlinear analysis, nonlinear functional analysis, partial differential equations, ordinary differential equations, differential geometry, and topology. For research mathematicians, physicists and graduate students.