0. Introduction. In his version of the operational calculus Mikusinski uses as a starting point the familiar theorem from algebra tlhat every integral domain can be embedded isomorphically in a field. He shows that the class of complex-valued continuous functions defined on [0, cO) forms an integral domain when addition and multiplication are taken to be pointwise addition and convolution of functions, respectively. The resulting field is called the field of Mikusinski operators. In the present paper we generalize Mikusinski's basic results by developing an operational calculus for strongly continuous functions defined on [0, cc) and having values in any Banach algebra with unit element. The Mikusinski operational calculus will then be a special case of our theory, obtained by taking the Banach algebra to be the complex numbers. The paper is divided into four parts. The first gives an algebraic construction; the second shows how the algebraic construction may be used to develop the operational calculus mentioned above; the third indicates some basic results which carry over from the (Mikusinski) operational calculus to this more general setting. Finally, in the last section, we generalize within the framework of the operational calculus a recent theorem of T. K. Boehme. We assume the reader is familiar with the basic notations and results in the books by Mikusinski [3] and Erdelyi [2].