The intrinsic normal cone

Let X be an algebraic stack in the sense of Deligne-Mumford. We construct a purely 0-dimensional algebraic stack over X (in the sense of Artin), the intrinsic normal cone C-x. The notion of (perfect) obstruction theory for X is introduced, and it is shown how to construct, given a perfect obstruction theory for X, a pure-dimensional virtual fundamental class in the Chow group of X. We then prove some properties of such classes, both in the absolute and in the relative context, Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.