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In this book, we concerned mainly with the evolute and the involute curves as well as their geometric properties in hyperbolic and de Sitter spaces. The evolute of a regular curve is a classical object from the viewpoint of differential geometry. The evolute and involute curves have the most important positions and applications in the study of design problems in spatial mechanisms and physics, kinematics, computer-aided design (CAD) and computer-aided geometric design (CAGD), it is one of the most important topics of differential geometry. Because of this, geometers have studied it in Euclidean, Minkowski, hyperbolic, and de Sitter spaces and they have investigated many properties for it. The evolute of a spherical curve is defined to be the locus of the center of its osculating spheres. Therefore, the evolute of a regular curve without inflection points is given by not only the locus of all its centers of curvature but also the envelope of its normal lines.