We investigate functions g(α,β;x) which are heavy-tailed Lévy stable probability distributions of index 0<α≤2 and appropriate asymmetry parameter β. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g(α,β;x) for all admissible rational values of α and β. We reproduce all the previously known cases and furnish distributions for values α and β which were previously inaccessible. We point out many instances of experimental and statistical data that could be described by our results.