  These are expanded notes from some talks given during the fall 2002, about
``homotopical algebraic geometry'' (HAG) with special emphasis on its
applications to ``derived algebraic geometry'' (DAG) and ``derived deformation
theory''.
  We use the general framework developed in Toen, Vezzosi, ``Homotopical
Algebraic Geometry I: Topos theory'', and in particular the notions of model
topology, model sites and stacks over them, in order to define various derived
moduli functors and study their geometric properties. We start by defining the
model category of D-stacks, with respect to an extension of the etale topology
to the category of non-positively graded commutative differential algebras, and
we show that its homotopy category contains interesting objects, such as
schemes, algebraic stacks, higher algebraic stacks, dg-schemes ... . We define
the notion of ``geometric D-stack'' and present some related geometric
constructions ($\mathcal{O}$-modules, perfect complexes, K-theory, derived
tangent stacks, cotangent complexes, various notion of smoothness ... .).
Finally, we define and study the derived moduli problems classifying local
systems on a topological space, vector bundles on a smooth projective variety,
and $A_{\infty}$-categorical structures. We state geometricity and smoothness
results for all of these examples.
