We will perform the calculation by two completely different methods, each of which has some advantages. In the first, we filter the algebra in question as in [2], and use a homogeneity argument to show that the resulting spectral sequence degenerates. From this point of view, the homology is seen to be a “semi-classical” invariant, since it is calculated at first order in Planck’s constant, as the homology of the differential forms on T ∗M with respect to the operator δ : Ω∗(T ∗M) → Ω∗−1(T ∗M) (defined by Koszul [14] and Brylinski [2]; see §1). The other method is a sheaf-theoretic calculation, modeled on Weil’s proof of de Rham’s Theorem, which uses a form of Poincare lemma for Hochschild homology of symbols. In the case of differential operators, this Poincare lemma states that the Hochschild homology of D(R), the algebra of differential operators on R, satisfies HH∗(D(R)) = H2n−∗(Rn,C). For pseudo-differential operators, the Poincare lemma is formulated for the sheaf ER of micro-differential operators on S∗M ×S, introduced in [16]; the copy of S comes from the fact that the sheaf ER lives on the cotangent bundle of a complexification of M . In the course of the paper, we also perform a number of other such calculations of Hochschild homology, for algebras of differential operators, operators with compact support, and formal deformations of C∞(X) for X a conic symplectic manifold.