In this doctoral thesis, the surfaces of constant mean curvature (CMC surfaces) and associated focal sets in Minkowski space are studied. Minkowski space is a special ambient space in which, with respect to defined pseudo-metric, we distinguish three types of vectors, curves and surfaces. In the first three chapters of the thesis, we give an overview of basic concepts and theorems in Minkowski space, local theory of curves and surfaces, as well as examples and classification of minimal, maximal and CMC surfaces in Minkowski space. In the chapter 4, the \(\alpha\)-evolute and its special case, the harmonic evolute of a surface, are analyzed. In the first part, we studied harmonic evolute of ruled, helicoidal and translation surfaces, respectively. Obtained results are analogous to the results in Euclidean case. In the second part, we studied the harmonic evolute of the quasi-umbilic surfaces. Contrary to the results in Euclidean case, the harmonic evolute of such surfaces is a curve. Moreover, it is proved that the harmonic evolute of a B-scroll of constant non-vanishing mean curvature, together with its base curve forms a null Bertrand pair. Furthermore, it is shown how to reconstruct B-scroll from a given harmonic evolute. At the end of the chapter, we studied properties of \(\alpha\)-evolute and the obtained results present significant extension of the results in Euclidean case. In the chapter 5, the cycloidal curves are considered. Their parametrization is presented, as well as shown that they can be obtained as projections of spherical generalized helices. By using the Björling formula, new examples of maximal surfaces are constructed, so-called cycloidal maximal surfaces. We derived their Weierstrass representations and proved that given surfaces are algebraic surfaces which admit rational parametrization.