In this article we describe a proposal to solve asymptotically free gauge theories, namely a method for doing computations in the strongly coupling regime of intermediate energy between the high energy perturbative region and the low energy region describe by the pseudo-Goldstone bosons of chiral symmetry breaking. The method looks for the most general scattering matrix, form factors and spectral densities that match the very low energy description in terms of a non-linear sigma model and the high energy in terms of perturbative QCD. As a result we obtain several resonances, the higher order coefficients of the chiral lagrangian, scattering lengths and various other low energy parameters. The basic input are the gauge theory parameters $N_c$, $N_f$, $\alpha_s$ and $m_\pi$, $f_\pi$.