  We completely characterise the bounded sets that arise as components of the
Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a
bounded domain is a Fatou component of some meromorphic function if and only if
it is regular. On the other hand, we prove that a planar continuum is a Julia
component of some meromorphic function if and only if it has empty interior. We
do so by constructing meromorphic functions with wandering continua using
approximation theory.
