Summary: In the Euclidean 3-space, there is a unique sphere for a curve \(\alpha:I\to \mathbb{R}^3\) such that the sphere touches \(\alpha\) at the third-order at \(\alpha (0)\). The intersection of the sphere with the osculating plane is a circle which touches \(\alpha\) at the second-order at \(\alpha(0)\) [5]. In this paper, the osculating sphere and the osculating circle of the curve are studied for each time-like, space-like and null (light-like) curves in semi-Riemannian spaces; \(\mathbb{R}^3_1\), \(\mathbb{R}^4_1\) and \(\mathbb{R}^4_2\). O’Neill, Barrett, Semi-Riemannian geometry. With applications to relativity. Pure Appl. Math., 103. New York-London etc.: Academic Press. XIII, 468 p. (1983; Zbl 0531.53051)].