An analysis based on 124000 selected tau pairs recorded by the ALEPH detector at LEP provides the vector (V) and axial-vector (A) spectral functions of hadronic tau decays together with their total widths. This allows the evaluation of finite energy chiral sum rules that are weighted integrals over the (VA) spectral functions. In addition, a precise measurement of alpha_s along with a determination of nonperturbative contributions at the tau mass scale is performed. The experimentally and theoretically most robust determination of alpha_s (M_tau^ 2) is obtained from the (V+ A) fit that yields alpha_s (M_tau^ 2)= 0.334+/-0.022 giving alpha_s (M_Z^ 2)= 0.1202+/-0.0027 after the extrapolation to the mass of the Z boson. The approach of the Operator Product Expansion (OPE) is tested experimentally studying the evolution of the tau hadronic widths to masses smaller than the tau mass.