  Separability for groups refers to the question which subsets of a group can
be detected in its finite quotients. Classically, separability is studied in
terms of which classes have a certain separability property, and this question
is related to algorithmic problems in groups such as the word problem. A more
recent perspective tries to study the order of the smallest finite quotient in
which one detects the subset under consideration depending on its complexity,
measured using the word norm on a finitely generated group. In this survey, we
present what is currently known in the field of effective separability and give
an overview of the open questions for several classes of groups.
