Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has been a longstanding problem to classify spaces of bounded rank four. Before our study, no non-classical example of such a space was known. We exhibit two non-classical examples of such spaces and give the full classification of basic spaces of bounded rank four. There are exactly four such up to isomorphism. We also take steps to bring together the methods of the linear algebra community and the algebraic geometry community used to study spaces of bounded rank.