  We propose that the statistics of the optimal tour in the planar random
Euclidean traveling salesman problem is conformally invariant on large scales.
This is exhibited in power-law behavior of the probabilities for the tour to
zigzag repeatedly between two regions, and in subleading corrections to the
length of the tour. The universality class should be the same as for dense
polymers and minimal spanning trees. The conjectures for the length of the tour
on a cylinder are tested numerically.
