This paper settles a problem raised at the end of the seventies by . Alperin [Al1], . Dade [Da] and . Carlson [Ca1], namely the classiﬁcation of torsion endo-trivial modules for a ﬁnite p-group over a ﬁeld of characteristic p. Our results also imply, at least when p is odd, the complete classiﬁcation of torsion endo-permutation modules. We refer to [CaTh] and [BoTh] for an overview of the problem and its importance in the representation theory of ﬁnite groups.