This talk will be concerned with compactness phenomena in set theory. Compactness is the phenomenon by which the local properties of a mathematical structure determine its global behaviour. This phenomenon is intrinsic to the architecture of the mathematical universe and manifests in various forms. Over the past fifty years, the study of compactness phenomena has been one of the flagships of research in set theory. This talk will present recent discoveries spanning classical themes like the tree property and stationary reflection while also forging new connections with other topics, such as Woodin's HOD Conjecture.