Ilı the Euclidean 3-space, t,here is a uııigue sphere for a curve o : / —» R^ such that the sphere touchc.s a at. the third order at ct(0). The iııtersectioıı of the splıere with oseulatiııg plaııe is a eircle which touches a at the secoud order at a(0) [5j. In this paper, the osculating sphere and the osculating circle of the curve are studied for each of timelike, spacelike and null (Ughtlike) curves in Semi-Riemannian Spaces; Rj, R| and R^.