Da Prato (Scuola Normale Superiore di Pisa) and Zabczyk (Polish Academy of Sciences, Warsaw) present the state of the rapidly evolving art of the theory of parabolic/elliptic equations in an infinite dimensional Hilbert space—theory relevant to functional analysis, mathematical physics, and applied mathematics. The theoretical section covers spaces of continuous functions and Sobolev spaces. Applications to control theory are then discussed. Appendices address interpolation spaces, null controllability, and semiconcave functions and Hamilton-Jacobi semigroups.