Random numbers are an essential tool in physics and information science. Quantum systems provide a particularly compelling source of such numbers because of the intrinsic randomness of quantum processes. An ideal quantum random number generator should be able to produce numbers that are algorithmically random (which is a strong, fundamental measure of randomness), and that are therefore incomputable (in contrast with the pseudo-random numbers generated by computers). But how can we test a generator for algorithmic randomness? And what does "intrinsic quantum randomness" mean anyway, and how can it be experimentally verified? In this talk, I first describe a photonic experiment we have done at the University of Portland that not only produces quantum random numbers, but also certifies the quantum randomness of the number-generating process through Bell's theorem. I will then describe our efforts to search for evidence of algorithmic randomness and incomputability in the output of both our own quantum random number generator and another generator realized at the Australian National University.