An abelian k-linear semisimple category having a finite number of simple objects, and endowed with a ribbon structure, is called premodular. It is modular (in the sense of Turaev) if the so-called S-matrix is invertible. A modular category defines invariants of 3-manifolds and a TQFT ([T]). When is it possible to construct a modularisation of a given premodular category, i.e. a functor to a modular category preserving the structures and ‘dominant' in a certain sense? It turns out (2.3) that this amounts essentially to making ‘transparent' objects trivial. We give a full answer to this problem in the case when k is a field of char. 0 (as well as partial answers in char. p): under a few obvious hypotheses, a premodular category admits a modularisation, which is unique (th. 3.1, and cor. 3.5 in char. 0) The proof relies on two main ingredients: a new and very simple criterion for the S-matrix to be invertible (1.1) and Deligne's internal characterization of tannakian categories in char. 0 [D]. When simple transparent objects are invertible, the criterion is simpler (4.2) and the modularisation can be described more explicitly (prop. 4.4). We conclude with two examples: the premodular categories associated with quantum and at roots of unity; in the first case, we obtain modular categories which were built independently by C. Blanchet [B]; in the second case, we obtain modularizations in all the cases where Y. Yokota [Y] found Reshetikhin-Turaev invariants of 3-manifolds, thereby improving as well as explaining Yokota's results.