  This is a survey paper on rainbow sets (another name for ``choice
functions''). The main theme is the distinction between two types of choice
functions: those having a large (in the sense of belonging to some specified
filter, namely closed up set of sets) image, and those that have a large domain
and small image, where ``smallness'' means belonging to some specified complex
(a closed-down set). The paper contains some new results: (1) theorems on
scrambled versions, in which the sets are re-shuffled before choosing the
rainbow set, and (2) results on weighted and cooperative versions - to be
defined below.
