This collection of invited lectures (at the Conference on Secondary Calculus and Cohomological Physics, Moscow) reflects the state-of-the-art in a new branch of mathematics and mathematical physics arising at the intersection of geometry of nonlinear differential equations, quantum field theory, and cohomological algebra. This is the first comprehensive and self-contained book on modern quantum field theory in the context of cohomological methods and the geometry of nonlinear PDEs. It features: an up-to-date and self-contained exposition of the newest results in cohomological aspects of quantum field theory and the geometry of PDEs; a new look at interrelations among cohomology theory, the geometry of PDEs, and field theory; and, application to Batalin-Vilkovisky formalism, BRST formalism, anomalies, and quantum dynamics.