We think of Frege's philosophy of arithmetic as constituted by a number of illuminating (and contentious) assertions about the natural numbers; that a statement of number is a statement about a concept, that numbers are logical objects, that truths about the arithmetical properties of numbers are logical truths. These are the central theses of his most accessible work, Die Grundlagen der Arithmetik. But Frege's writings on arithmetic, both before and after the Grundlagen, include discussion of the real and, on occasions, the complex numbers as well. My primary aim here is to provide an analysis, largely historical, of Frege's theory of real numbers. In order to see how the theory developed and to understand its ramifications in other areas of Frege's philosophy I shall embed the discussion of real numbers in a more general framework which will include Frege's concept of a course of values — for this is central to his logical construction of number theory and analysis — and his views on the relation between arithmetic and geometry. This should lead to a better appreciation of some of the changes which his views underwent during fifty years of intense occupation with the nature of mathematics.