We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds S_{\epsilon,\zeta} under more restrictive assumptions on the linear part of the slow equation. The second parameter \zeta does not appear in the finite-dimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which S_{\epsilon,\zeta} does not depend on \zeta . Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.