  We provide a classification of minimal sets of homeomorphisms of the
two-torus, in terms of the structure of their complement. We show that this
structure is exactly one of the following types: (1) a disjoint union of
topological disks, or (2) a disjoint union of essential annuli and topological
disks, or (3) a disjoint union of one doubly essential component and bounded
topological disks. Periodic bounded disks can only occur in type 3. This result
provides a framework for more detailed investigations, and additional
information on the torus homeomorphism allows to draw further conclusions. In
the non-wandering case, the classification can be significantly strengthened
and we obtain that a minimal set other than the whole torus is either a
periodic orbit, or the orbit of a periodic circloid, or the extension of a
Cantor set. Further special cases are given by torus homeomorphisms homotopic
to an Anosov, in which types 1 and 2 cannot occur, and the same holds for
homeomorphisms homotopic to the identity with a rotation set which has
non-empty interior. If a non-wandering torus homeomorphism has a unique and
totally irrational rotation vector, then any minimal set other than the whole
torus has to be the extension of a Cantor set.
