The main aim of my talk will be to explain why a geometer should ever care about a (derived) categorical approach to the subject. This means I shall try to reduce to a minimum any homological/categorical algebra and ignore technicalities with the derived formalism. Instead I would like to explain the first (chronologically) example of a derived equivalence between non-isomorphic varieties, namely Mukai’s derived equivalence between an abelian variety and its dual. Time-permitting I hope to explain how this gives rise to some SL_2(Z)-like action on the derived category of an elliptic curve.