We present a general theory of payment systems that is capable of describing both traditional and electronic forms of payment. Starting from the three basic functions of money and general non-functional requirements, we derive the necessary and sufficient properties of technical implementations of money and payments. We describe possible scalable implementations of e-money schemes based on a general description of their data structures (money distributions) and payments. We define the notion of bill scheme, in which the value units are bills with invariant values, and show that only the bill scheme allows for scalable and practically efficient implementations through decomposition, where the components have to process a considerably smaller amount of data and a number of payment requests, compared to the whole system.