The generalized Lagrange probability distributions include the Borel-Tanner distribution, Haight's distribution, the Poisson-Poisson distribution and Consul's distribution, to name a few. We introduce two universally applicable random variate generators for this family of distributions. In the branching process method, we produce the generation sizes in a Galton-Watson branching process. In the uniform bounding method, we employ the rejection method based upon a simple probability inequality that is valid for id members in a given subfamily.