For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category $\\sA$ whose radical $\\sR$ is locally nilpotent and such that $\\bar \\sA:=\\sA/\\sR$ is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection $\\sA\\to \\bar\\sA$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when $\\sA$ has a monoidal structure for which $\\sR$ is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson-Morozov theorem: the existence of a pro-reductive envelope $\\Pred(G)$ associated to any affine group scheme $G$ over $K$. Other applications are given in this paper as well as in a forthcoming one on motives.