  We construct virtual fundamental classes for dg-manifolds whose tangent
sheaves have cohomology only in degrees 0 and 1. This condition is analogous to
the existence of a perfect obstruction theory in the approach of
Behrend-Fantechi [BF] or Li-Tian [LT]. Our class is initially defined in
K-theory, as the class of the structure sheaf of the dg-manifold.
  We compare our construction with that of [BF] as well as with the original
proposal of Kontsevich. We prove a Riemann-Roch type result for dg-manifolds
which involves integration over the virtual class. We prove a localization
theorem for our virtual classes. We also associate to any dg-manifold of our
type a cobordism class of almost complex (smooth) manifolds. This supports the
intuition that working with dg-manifolds is the correct algebro-geometric
replacement of the analytic technique of "deforming to transversal
intersection".
