Numerous recent experiments, including nuclear magnetic resonance, specific heat, and upper critical field measurements, have provided concurring evidence of spontaneously broken rotational symmetry in the superconducting state of the doped topological insulator CuxBi2Se3. Such state can be called a nematic superconductor, in analogy with liquid crystals. In this talk, I will demonstrate that the experimental results provide strong support for an unconventional odd-parity pairing, leading to a realization of topological superconductivity. I will present the phenomenological Ginzburg-Landau theory of such superconductors, which, combined with the symmetry-based microscopic analysis, allows to explain existing experiments and make new predictions. Finally, I will discuss the microscopic mechanism for nematic superconductivity from density wave fluctuations, which may explain the very recent experiments on twisted bilayer graphene.