  We define and study an equivariant version of Farber's topological complexity
for spaces with a given compact group action. This is a special case of the
equivariant sectional category of an equivariant map, also defined in this
paper. The relationship of these invariants with the equivariant
Lusternik-Schnirelmann category is given. Several examples and computations
serve to highlight the similarities and differences with the non-equivariant
case. We also indicate how the equivariant topological complexity can be used
to give estimates of the non-equivariant topological complexity.
