This paper presents a technique for efficiently generating random numbers from a given probability distribution. This is achieved by using a generic hardware architecture, which transforms uniform random numbers according to a distribution mapping stored in RAM, and a software approximation generator that creates distribution mappings for any given target distribution. This technique has many features not found in current non-uniform random number generators, such as the ability to adjust the target distribution while the generator is running, per-cycle switching between distributions, and the ability to generate distributions with discontinuities in the Probability Density Function.