This book is based on a course given by the author at the Institut de Mathématiques de Jussieu in 2004-2005. The topic treated in the book is the derived category of coherent sheaves on a smooth projective variety. In the last couple of years, the Kontsevich homological mirror symmetry has revived interest in these questions by proposing an equivalence of the derived category of coherent sheaves of certain projective varieties with the Fukaya category associated to the symplectic geometry of the mirror variety. It is this point of view that motivates and in some sense explains many of the central results as well as the open problems in this area. The most prominent example of this equivalence was observed by Mukai in the very first paper on the subject. He showed that the Poincaré bundle induces an equivalence between the derived category of an Abelian variety A and the derived category of its dual Ă, although Ă is generally not isomorphic to A. These results naturally led to the question, ‘under what conditions do two smooth projective varieties give rise to equivalent derived categories?’, which is the central theme of the book.