The locus of centers of osculating circles is called the evolute of a curve. The evolute is also the envelope of the family of normal lines to the curve. From 1665, when the mathematician Huygens introduced the notion of the evolute of a conic in the Euclidean plane, to the present day numerous papers have been written. In this doctoral thesis, the properties of evolutes are studied with emphasis on their characteristics related to Plücker formulas (order, class, number of double points/tangents, number of cups, inflection points/lines) in seven out of a total of nine projective-metric planes. The thesis is divided into five chapters in which an extensive review about basic concepts and properties of the evolute of a conic is made in the Euclidean plane, pseudo-Euclidean plane, quasi-hyperbolic plane, quasi-elliptic plane and projective extended hyperbolic plane. The thesis studies in detail how the position of the conic towards the absolute figure affects the order and class of its evolute. The syntehetic and analytical methods have been used in the studies. The computer programs used in the research are The Geometer’s Sketchpad, Geogebra, Wolfram Mathematica and Demos. The main aim of this thesis is to systematize the facts related to prominent projective-metric planes and scientific contribution to the classification of evolutes and their constructive processing.