For a large class of linear continuous-time systems including delay-differential systems, an algebraic theory is presented in terms of Noetherian operator rings generated from a finite number of elements belonging to a convolution algebra of distributions. The external behavior of these systems is given by a finite set of input/output (convolution) operator equations which are solved in a novel manner by constructing an operational transfer function matrix and then applying an extension of the Mikusinski operational calculus. After the formulation of an internal representation consisting of a finite set of scalar operational-differential equations, the problem of realizing an operational transfer function matrix by such an internal description is considered. Results on the existence and construction of realizations are given.