Turbulent drag reduction by dilute addition of high polymers is studied by considering local stretching of the molecular structure of a polymer by small-scale turbulent motions in the region very close to the wall. The stretching process is assumed to restructure turbulence at small scales by forcing these to satisfy local axisymmetry with invariance under rotation about the axis aligned with the main flow. It can be shown analytically that kinematic constraints imposed by local axisymmetry force turbulence near the wall to tend towards the one-component state and when turbulence reaches this limiting state it must be entirely suppressed across the viscous sublayer. For the limiting state of wall turbulence, the statistical dynamics of the turbulent stresses, constructed by combining the two-point correlation technique and invariant theory, suggest that turbulent drag reduction by homogeneously distributed high polymers, cast into the functional space which emphasizes the anisotropy of turbulence, resembles the process of reverse transition from the turbulent state towards the laminar flow state. These findings are supported by results of direct numerical simulations of wall-bounded turbulent flows of Newtonian and non-Newtonian fluids and by experiments carried out, under well-controlled laboratory conditions, in a refractive index-matched pipe flow facility using state-of-the art laser-Doppler anemometry. Theoretical considerations based on the elastic behavior of a polymer and spatial intermittency of turbulence at small scales enabled quantitative estimates to be made for the relaxation time of a polymer and its concentration that ensure maximum drag reduction in turbulent pipe flows, and it is shown that predictions based on these are in very good agreement with available experimental data.