  While the dynamics of transcendental entire functions in periodic Fatou
components and in multiply connected wandering domains are well understood, the
dynamics in simply connected wandering domains have so far eluded
classification. We give a detailed classification of the dynamics in such
wandering domains in terms of the hyperbolic distances between iterates and
also in terms of the behaviour of orbits in relation to the boundaries of the
wandering domains. In establishing these classifications, we obtain new results
of wider interest concerning non-autonomous forward dynamical systems of
holomorphic self maps of the unit disk. We also develop a new general technique
for constructing examples of bounded, simply connected wandering domains with
prescribed internal dynamics, and a criterion to ensure that the resulting
boundaries are Jordan curves. Using this technique, based on approximation
theory, we show that all of the nine possible types of simply connected
wandering domain resulting from our classifications are indeed realizable.
