Coherent interfaces between optical photons and long-lived matter qubits form a key resource for a broad range of quantum technologies. Cavity quantum electrodynamics (cQED) offers a route to achieve such an interface by enhancing interactions between cavity-confined photons and individual emitters. Over the last two decades, a promising new class of emitters based on defect centers in diamond have emerged, combining long spin coherence times with atom-like optical transitions. More recently, advances in optical resonator technologies have made it feasible to realize cQED in diamond. This article reviews progress towards coupling color centers in diamond to optical resonators, focusing on approaches compatible with quantum networks. We consider the challenges for cQED with solid-state emitters and introduce the relevant properties of diamond defect centers before examining two qualitatively different resonator designs: micron-scale Fabry-Perot cavities and diamond nanophotonic cavities. For each approach, we examine the underlying theory and fabrication, discuss strengths and outstanding challenges, and highlight state-of-the-art experiments.