Abstract: "Derived algebraic geometry, developped <br /> in collaboration with G. Vezzosi, is a general theory <br /> whose purpose is to "do geometry over the category <br /> of complexes", and which is essentially obtained <br /> by replacing commutative rings by commutative <br /> differential graded algebras. The purpose of this talk <br /> is to present informally the general concepts of <br /> derived algebraic geometry, explaining how it provides <br /> a global counter-part to the "derived deformation theory" <br /> of Drinfeld, Kontsevich and others. In a second part of the <br /> talk I will present recent developments, <br /> including the construction several "derived" moduli spaces, <br /> application to virtual fundamental classes, <br /> and the construction of moduli for complexes of sheaves on a scheme."