  The methods of conformal field theory are used to compute the crossing
probabilities between segments of the boundary of a compact two-dimensional
region at the percolation threshold. These probabilities are shown to be
invariant not only under changes of scale, but also under mappings of the
region which are conformal in the interior and continuous on the boundary. This
is a larger invariance than that expected for generic critical systems.
Specific predictions are presented for the crossing probability between
opposite sides of a rectangle, and are compared with recent numerical work. The
agreement is excellent.
