  Let f be a real entire function whose set S(f) of singular values is real and
bounded. We show that, if f satisfies a certain function-theoretic condition
(the "sector condition"), then $f$ has no wandering domains. Our result
includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real
constant and {\lambda} is positive.
  We also show the absence of wandering domains for certain non-real entire
functions for which S(f) is bounded and the iterates of f tend to infinity
uniformly on S(f).
  As a special case of our theorem, we give a short, elementary and
non-technical proof that the Julia set of the complex exponential map f(z)=e^z
is the entire complex plane.
  Furthermore, we apply similar methods to extend a result of Bergweiler,
concerning Baker domains of entire functions and their relation to the
postsingular set, to the case of meromorphic functions.
