%0 Journal Article %T Cavity quantum electrodynamics with three-dimensional photonic bandgap crystals %A W. L. Vos %A L. A. Woldering %J Physics %D 2015 %I arXiv %X This paper gives an overview of recent work on three-dimensional (3D) photonic crystals with a "full and complete" 3D photonic band gap. We review five main aspects: 1) spontaneous emission inhibition, 2) spatial localization of light within a tiny nanoscale volume (aka "a nanobox for light"), 3) the introduction of a gain medium leading to thresholdless lasers, 4) breaking of the weak-coupling approximation of cavity QED, both in the frequency and in the time-domain, 5) decoherence, in particular the shielding of vacuum fluctuations by a 3D photonic bandgap. In addition, we list and evaluate all known photonic crystal structures with a demonstrated 3D band gap. %U http://arxiv.org/abs/1504.06803v1