The problem of transforming nonlinear control systems into input-output prime forms is dealt with, using state space. static state feedback, and also output transformations. Necessary and sufficient geometric conditions for the solvability of this problem are obtained. The results obtained generalize well-known results both on feedback linearization as well as input-output decoupling of nonlinear systems. It turns out that, from a computational point of view, the output space transformation is the crucial step, that is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold.