pProviding an introduction to stochastic optimal control in in64257nite dimension, this book gives a complete account of the theory of second-order HJB equations in in64257nite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results e.g. the dynamic programming principle with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in in64257nite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochasti