In nature, many of the smallest-scale events come down to chance: will an atom undergo a radioactive decay in a given time, or won't it? Will a  single photon be reflected at a window, or pass through? Quantum theory insists that the outcomes of these events cannot be known in advance: they are random. But is this randomness an intrinsic characteristic of nature, or does it arise merely from our ignorance about what is "really going on" in quantum systems? The question of whether or not quantum mechanics is a complete description of nature was first raised by Einstein, Podolsky, and Rosen in 1935. Later, John Bell discovered a mathematical inequality that let us test, experimentally, whether it is even possible to construct more complete "local-realistic" theories to replace quantum mechanics. I will discuss the Einstein Podolsky-Rosen paradox, the Bell Inequalities, and several of the landmark experiments that have been performed in the past 30 years to test them.