The present book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of algebraic curves. It is based on the lecture courses given by Edward Frenkel at Harvard University in the Fall of 1996 and at UC Berkeley in the Spring of 1999. The notes of these lectures were taken by David Ben-Zvi. They were subsequently improved and expanded by both authors. The main goal of this book is to introduce the concept of vertex algebra in a coordinate-independent way, and to define the spaces of conformal blocks attached to an arbitrary vertex algebra and a smooth algebraic curve, possibly equipped with some extra geometric data. From this point of view vertex algebras appear as the algebraic objects that encode the local geometric structure of various moduli spaces associated with algebraic curves.