This treatment of differential geometry and the mathematics required for general relativity makes the subject matter accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. To maintain the flow of exposition and to develop experience in the reader, the technicalities of many proofs are in the form of carefully programmed exercises rather than endless pages of displayed equations. In this way, the reader is guided through actually doing computations instead of being allowed to skim over a page of ready worked examples. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists.