This book originates from a graduate course in algebraic geometry held in the summer of 2010. I introduced many fundamental techniques in algebraic geometry and explained in detail how they are applied to K3 surfaces. The diversity of the theory of K3 surfaces, touching upon so many topics in both algebraic geometry and other areas, including arithmetic, complex and differential geometry, homological algebra, and even mathematical physics, is fascinating. I hoped to convey some of this fascination and, at the same time, to demonstrate how the various aspects – ranging from Hodge theory to moduli spaces to derived categories – come together in a meaningful way when studied for K3 surfaces.

Over time, the original lecture notes have grown. They now cover large parts, but by no means all, of the theory of K3 surfaces. As the notes made available online appeared to be useful, it seemed worthwhile to turn them into this book. I hope it will serve as an introduction to the subject as well as a guide to the vast literature. The balance between these two goals turned out to be difficult to achieve. Some chapters are more or less self-contained, while others are certainly not and rather are meant as an invitation to consult the original sources.

Each chapter is devoted to a different topic and presents the relevant theory in a condensed form, accompanied by extensive references to the original articles and to the relevant textbooks. Sometimes the text can be read as a survey, while other times technical aspects particular to K3 surfaces are discussed in detail. Often, I try to give ad hoc arguments that work only for K3 surfaces, though a more powerful general theory may be available. The aim is to allow the chapters to be read independently of each other, encouraging a non-linear reading. Although this goal was not fully achieved, I hope I at least made it easy to navigate between the chapters. Also, I have not hesitated to revisit some aspects to emphasize different angles or give more details.

The choice of topics is a personal one and I am aware of many omissions.