A synthesis of the classic Double Fourier sphere method and new ideas in low rank approximation is used to construct approximations to functions in polar and spherical geometries. This approach preserves bi-periodicity for functions on the sphere, overcomes issues associated with artificial singularities, enables the use of fast, FFT-based algorithms, and is near-optimal in its underlying interpolation scheme. It has been used to develop a suite of fast, scalable algorithms for numerical computing with functions on the surface of the sphere and on the disk, and these have been implemented in the Chebfun computing system.