This is a book on the interplay between linear and semi-linear, elliptic and parabolic differential equations on one side, and the theory of diffusions and superdiffusions on the other side. Superdiffusion can be viewed as a diffusion of a “cloud” of particles, obeying suitable rules of branching (i.e. birth and death of the particles). There are integral formulas resembling - and greatly generalizing - the classical Feynman-Kac path integrals. The author is a leading expert in the field; he played a key role in the development of probability theory since the fifties. The book gives a detailed treatment of the progress achieved by him and his collaborators in the last 12 years. An intuitive explanation of some of the main ideas is given in the introduction. Part 1 (Parabolic equations and branching exit Markov systems) contains chapters on linear PE and diffusions, BEM systems, superprocesses, semilinear parabolic equations and superdiffusions. Part 2 (Elliptic equations and diffusions) contains chapters on linear EE and diffusions, positive harmonic functions, moderate solutions of the equation Lu=ψ(u), stochastic boundary values of solutions, rough trace, fine trace, the Martin capacity, null sets and polar sets. In the appendices, the reader can find facts on martingales and elliptic differential equations. The book combines both probabilistic and analytic tools with a very high skill; it summarizes the results achieved in an important area, which progressed significantly in recent years.