A class of Mikusiiski-type operators in several variables, called ncocontin- uous operators, is studied. These particular operators are closely affiliated with Schwartz distributions on Rk and share certain continuity properties with them. This affiliation is first of all revealed through a common algebraic view of operators and distributions as homomorphic mappings and a new representation theory, and is then characterized in terms of continuity properties of the mappings. The traditional procedures of the operational calculus apply to the class of neocontinuous operators. Moreover, the somewhat vague association of operational and distributional solutions of partial differen- tial equations is replaced by the decisive representation concept, thus illustrating the appropriateness of the study of neocontinuous operators. 1. Introduction. In this paper we investigate a class of Mikusiniski-type operators in several variables together with a class of Schwartz distributions. Using an algebraic view (9) of operators and distributions as homomorphic mappings, we consider those operators which, when restricted to suitable domains, are continuous and agree with one or more distributions. These operators are said to represent the distributions and are called neocontinuous operators. The concept of representation referred to here is considerably more general than the traditional one used to identify certain operators and distribu- tions having suitably bounded supports, and has been formulated only recently (10) in an attempt to treat partial differential operators by algebraic methods. Thus the class of neocontinuous operators is quite large and can be expected to play a central role in the operational calculus in several variables. Our attention was first drawn to this class of neocontinuous operators by the familiar proof of the Malgrange-Ehrenpreis theorem on the existence of funda- mental functions for (partial) differential operators (with constant coefficients). In this proof, L2-inequalities for polynomials are used to define a continuous linear form (which inverts the operator) on a subspace of test functions and which is then extended to all test functions by the Hahn-Banach theorem. A version of this technique is employed here to characterize the operators we wish to study and provides for a more intimate affiliation with distributions than heretofore considered in the Mikusin'ski operational calculus.