Entanglement is the key resource that allows quantum protocols for simulation, computation, and measurement to outperform their classical counterparts. Optical cavities facilitate efficient generation of entanglement by using light to mediate long-range interactions among thousands of atoms. In our present work, we program the spatial structure of entanglement by using local spin rotations to break the all-to-all symmetry of cavity-mediated interactions. Illustrating this programmability, we generate “graph states” of up to four atomic ensembles where the eponymous graph specifies an arbitrary entanglement structure. We have further demonstrated programmable interactions among up to 18 ensembles of atoms, realizing correlation structures that map to exotic geometries, including a Möbius ladder. These results show versatile control over quantum dynamics, with direct applications to multi-mode quantum measurement protocols and the simulation of frustrated spin systems.