  Profinite semigroups are a generalization of finite semigroups that come
about naturally when one is interested in considering free structures with
respect to classes of finite semigroups. They also appear naturally through
dualization of Boolean algebras of regular languages. The additional structure
is given by a compact zero-dimensional topology. Profinite topologies may also
be considered on arbitrary abstract semigroups by taking the initial topology
for homomorphisms into finite semigroups. This text is the proposed chapter of
the Handdbook of Automata Theory dedicated to these topics. The general theory
is formulated in the setting of universal algebra because it is mostly
independent of specific properties of semigroups and more general algebras
naturally appear in this context. In the case of semigroups, particular
attention is devoted to solvability of systems of equations with respect to a
pseudovariety, which is relevant for solving membership problems for
pseudovarieties. Focus is also given to relatively free profinite semigroups
per se, specially "large" ones, stressing connections with symbolic dynamics
that bring light to their structure.
