Stochastic processes in infinite dimensional state spaces provide a mathematical description of various phenomena in physics, population biology, finance, and other fields of science. Several aspects of these processes have been studied in this thesis by means of new analytic methods. Firstly, Kolmogorov equations associated with a class of infinite dimensional diffusions are considered. In this thesis the problem of obtaining domain characterisations for the generators is solved in the infinite dimensional non-symmetric case. Secondly, Fokker-Planck equations associated with infinite dimensional diffusions are studied. It is shown that these equations can be interpreted as gradient flows in the space of probability measures over an infinite dimensional Banach space. Finally, the thesis deals with Malliavin calculus, a very useful calculus for obtaining regularity results for stochastic (partial) differential equations. So far the theory has been restricted to Hilbert spaces, but it is demonstrated in this thesis that the theory extends to more general Banach spaces.