  We address the construction of stable random matrix ensembles as the
generalization of the stable random variables (Levy distributions). With a
simple method we derive the Cauchy case, which is known to have remarkable
properties. These properties allow for such an intuitive method -that relies on
taking traces- to hold. Approximate but general results regarding the other
distributions are derived as well. Some of the special properties of these
ensembles are evidenced by showing partial failure of mean-field approaches. To
conclude, we compute the confining potential that gives a Gaussian density of
states in the limit of large matrices. The result is an hypergeometric
function, in contrast with the simplicity of the Cauchy case.
