Photons do not interact strongly in nature, and have thus been relegated to a role as a tool rather than an object of study in condensed matter physics. However, in cavity quantum electrodynamics, the strong interaction of light with a single atom can lead to strong atom-mediated photon-photon interactions, even when the light and atomic transitions are not resonant. Recent theoretical proposals have predicted phase transitions in arrays of these cavities, demonstrating that complex matter-like phenomena can emerge from a sea of interacting photons. I will present our recent measurements demonstrating strong photon-photon interactions in superconducting cavity QED. Here, we observe a photon blockade effect, where the presence of one photon blocks further transmission through the cavity when probing the system with a method in direct analogy to electron transport measurements in quantum dots. I will also present measurements of cavity arrays to understand disorder, and discuss prospects for observing phase transitions and effects of broken time reversal symmetry in these arrays.