This paper studies in great detail a family of supersymmetric Wilson loop
operators in N=4 supersymmetric Yang-Mills theory we have recently found. For a
generic curve on an S^3 in space-time the loops preserve two supercharges but
we will also study special cases which preserve 4, 8 and 16 supercharges. For
certain loops we find the string theory dual explicitly and for the general
case we show that string solutions satisfy a first order differential equation.
This equation expresses the fact that the strings are pseudo-holomorphic with
respect to a novel almost complex structure we construct on AdS_4 x S^2. We
then discuss loops restricted to S^2 and provide evidence that they can be
calculated in terms of similar observables in purely bosonic YM in two
dimensions on the sphere.Comment: Latex, 84 pages, 4 figures. v2: minor changes, references added; to
appear in JHE
