  Given CW complexes X and Y, let map(X,Y) denote the space of continuous
functions from X to Y with the compact open topology. The space map(X,Y) need
not have the homotopy type of a CW complex. Here the results of an extensive
investigation of various necessary and various sufficient conditions for
map(X,Y) to have the homotopy type of a CW complex are exhibited. The results
extend all previously known results on this topic. Moreover, appropriate
converses are given for the previously known sufficient conditions.
  It is shown that this difficult question is related to well known problems in
algebraic topology. For example, the geometric Moore conjecture (asserting that
a simply connected finite complex admits an eventual geometric exponent at any
prime if and only if it is elliptic) can be restated in terms of CW homotopy
type of certain function spaces.
  Spaces of maps between CW complexes are a particular case of inverse limits
of systems whose bonds are Hurewicz fibrations between spaces of CW homotopy
type. Related problems concerning CW homotopy type of the limit space of such a
system are also studied. In particular, an almost complete solution to a well
known problem concerning towers of fibrations is presented.
