  We study Wilson-'t Hooft loop operators in a class of N=2 superconformal
field theories recently introduced by Gaiotto. In the case that the gauge group
is a product of SU(2) groups, we classify all possible loop operators in terms
of their electric and magnetic charges subject to the Dirac quantization
condition. We then show that this precisely matches Dehn's classification of
homotopy classes of non-self-intersecting curves on an associated Riemann
surface--the same surface which characterizes the gauge theory. Our analysis
provides an explicit prediction for the action of S-duality on loop operators
in these theories which we check against the known duality transformation in
several examples.
