Mukai models for K3 surfaces in low degrees

The thesis consists of a detailed study of the classification of (polarised) K3 surfaces, which are among the most studied families of varieties in algebraic geometry since their introduction by André Weil in the 50s. The classification is made according to an algebraic invariant called genus, or equivalently, according to the degree. We focus on the innovations and techniques arising from the work of Mukai from the second part of the 80s to these days. We present every general model of K3 surface known at the present time along with some examples and constructions related to it. In doing so, we state and prove the main result by Mukai, the so-called vector bundle method. Vector bundles on K3 surfaces and on Fano varieties play a fundamental role in the study of the models. Moreover, the structure of homogeneous space carried by grassmannians and other Fano varieties allows us to use a combinatoric approach for the study of vector bundles and other algebraic invariants of these varieties. The study of the classification is strictly linked to the study of the moduli spaces of polarised K3 surfaces in each genus, which is not completely understood. However, we know from a result of Gritsenko, Hulek and Sankaran that it is not possible to find a general model for a high genus. Anyway, for the previous values of the genus where it is theoretically possible to find the model, we still do not have an explicit description of the model for each genus. Hence, the classification is incomplete at the present time. At the end of the work, we give some ideas for why it is so difficult to find new models and what are the possible different directions for the study of these surfaces.