Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Lévy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Lévy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Lévy tail distributions are characterized by a different novel form of microscopic universality.