  We survey recent results about composition operators induced by analytic
self-maps of the unit disk in the complex plane on various Banach spaces of
analytic functions taking values in infinite-dimensional Banach spaces. We
mostly concentrate on the research line into qualitative properties such as
weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in
several other papers. We discuss composition operators on strong, respectively
weak, spaces of vector-valued analytic functions, as well as between weak and
strong spaces. As concrete examples, we review more carefully and present some
new observations in the cases of vector-valued Hardy and BMOA spaces, though
the study of composition operators has been extended to a wide range of spaces
of vector-valued analytic functions, including spaces defined on other domains.
Several open problems are stated.
