Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding two-dimensional topological eld theories. Our approach is to use the formalism of operads. Operads can be de ned in any symmetric monoidal category, although we will mainly be concerned with dg-operads (di erential graded operads), that is, operads in the category of chain complexes with monoidal structure de ned by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative | we give a precise de nition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a non-linear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the k-th tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or a-algebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)

. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the p-local homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...