"K3 surfaces, which were formally defined by André Weil in 1958, turned out to be among the most fascinating mathematical objects and had considerable impact outside algebraic geometry. For example, the differential manifold underlying a K3 surface has the kind of intersection pairing that is responsible for its remarkable role in the development of 4-dimensional topology in the 1980s. But such surfaces also admit a hyperkähler metric, and this largely accounts for their importance in complex differential geometry and string theory. Today the period map for K3 surfaces is well-understood, but deeper algebro-geometric properties, including those of a motivic and arithmetic nature, are not. Indeed, that these areas are currently the subject of intense research has increased the need for an up-to-date introduction to the recent literature. This systematic exposition by a renowned expert meets that need wonderfully." Eduard Looijenga, Tsinghua University, Beijing and Universiteit Utrecht, The Netherlands.