  In this paper we present a family of supersymmetric Wilson loops of N=4
supersymmetric Yang-Mills theory in Minkowski space. Our examples focus on
curves restricted to hyperbolic submanifolds, H_3 and H_2, of space-time.
Generically they preserve two supercharges, but in special cases more,
including a case which has not been discussed before, of the hyperbolic line,
conformal to the straight line and circle, which is half-BPS. We discuss some
general properties of these Wilson loops and their string duals and study
special examples in more detail. Generically the string duals propagate on a
complexification of AdS_5 x S^5 and in some specific examples the compact
sphere is effectively replaced by a de-Sitter space.
