This equation is a close cousin of the Fermat equation xn + yn = zn. As for the Fermat equation, one is interested in determining all primitive nontrivial integral solutions. Here nontrivial means that none of the variables vanishes and primitive means that x, y and z have no common prime divisor.
By a general result on generalized Fermat equations, one knows that there exist only finitely many of these solutions. Five such solutions are known; some of them are pretty large, for example