Let \(X\) be a reduced simplicial set, \(GX\) its loop complex à la Kan and \(F\) a simplicial set on which \(GX\) acts from the left. Then the twisted product \(X \widetilde{\times} F\) is defined; it is the simplicial analogue of a topological fibre space. A twisting cochain is a map \(\tau : C^N_* (X) \to C^N_{* - 1} (GX)\) satisfying certain algebraic properties \((C^N_*\) denoting the complex of normalized chains). It is used to define a twisted differential \(d_\tau\) on the tensor product \(C^N_*(X) \otimes C_*(F)\) such that this chain complex is chain homotopy equivalent to \(C_*(X \widetilde{\times} F)\). Using the technique of acyclic models, the authors construct a twisting cochain which is natural in \(X\) and is compatible with the Eilenberg-MacLane transformation \(C^N_*(X \times Y) \overset \sim {} C^N_*(X) \otimes C^N_*(Y)\).