The memoir presents a systematic study of rational $S^1$-equivariant cohomology theories, and a complete algebraic model for them. It provides a classification of such cohomology theories in simple algebraic terms and a practical means of calculation. The power of the model is illustrated by analysis of the Segal conjecture, the behavior of the Atiyah-Hirzebruch spectral sequence, the structure of $S^1$-equivariant $K$-theory, and the rational behavior of cyclotomic spectra and the topological cyclic homology construction.