This is a review article of mathematical results in cosmology, written in honor of Yvonne Choquet-Bruhat’s 100th birthday. It starts with a brief description of some of the essential questions: strong cosmic censorship; the relation between the future asymptotics and geometrization in the vacuum setting; the cosmic no-hair conjecture; and the BKL-proposal. It then turns to results, starting with ones obtained in situations with symmetry. It continues with a review of future global non-linear stability results, stable big bang formation results, results concerning solutions to linear equations on cosmological backgrounds and numerical results. The article contains a substantial, but very incomplete, list of references to the literature.