A structure has a (finite-string) automatic presentation if the
elements of its domain can be named by finite strings in such a way
that the coded domain and the coded atomic operations are recognised
by synchronous multitape automata. Consequently, every structure with
an automatic presentation has a decidable first-order theory. The
problems surveyed here include the classification of classes of
structures with automatic presentations, the complexity of the
isomorphism problem, and the relationship between definability and
recognisability.