  Although detailed descriptions of the possible types of behaviour inside
periodic Fatou components have been known for over 100 years, a classification
of wandering domains has only recently been given. Recently, simply connected
wandering domains were classified into nine possible types and examples of
escaping wandering domains of each of these types were constructed. Here we
consider the case of oscillating wandering domains, for which only six of these
types are possible. We use a new technique based on approximation theory to
construct examples of all six types of oscillating simply connected wandering
domains. This requires delicate arguments since oscillating wandering domains
return infinitely often to a bounded part of the plane. Our technique is
inspired by that used by Eremenko and Lyubich to construct the first example of
an oscillating wandering domain, but with considerable refinements which enable
us to show that the wandering domains are bounded, to specify the degree of the
mappings between wandering domains and to give precise descriptions of the
dynamical behaviour of these mappings.
