  Classification is an important goal in many branches of mathematics. The idea
is to describe the members of some class of mathematical objects, up to
isomorphism or other important equivalence in terms of relatively simple
invariants. Where this is impossible, it is useful to have concrete results
saying so. In model theory and descriptive set theory, there is a large body of
work, showing that certain classes of mathematical structures admit
classification, while others do not. In the present paper, we describe some
recent work on classification in computable structure theory.
