Title: Generating Realistic 3D Molecules with an Equivariant Conditional Likelihood Model. Abstract: The number of drug-like molecules that could potentially exist is thought to be above $10^{33}$, precluding exhaustive computational or experimental screens for molecules with desirable pharmaceutical properties. Machine learning models that can propose novel molecules with specific characteristics are powerful new tools to break through the intractability of searching chemical space. Most of these models generate molecular graphs—representations that describe the topology of covalently bonded atoms in a molecule—because the bonding information in the graphs is required for many downstream applications, such as virtual screening and molecular dynamics simulation. These models, however, do not themselves generate 3D coordinates for the atoms within a molecule (which are also required for these applications), and thus they cannot easily incorporate information about 3D geometry when optimizing molecular properties. In this paper, we present GEN3D, a model that concurrently generates molecular graphs and 3D geometries, and is equivariant to rotations, translations, and atom permutations. The model extends a partially generated molecule by computing a conditional distribution over atom types, bonds, and spatial locations, and then sampling from that distribution to update the molecular graph and geometries, one atom at a time. We found that GEN3D proposes molecules that have much higher rates of chemical validity, and much better atom-distance distributions, than those generated with previous models. In addition, we validated our model’s geometric accuracy by forcing it to predict geometries for benchmark molecular graph inputs, and found that it also advances the state of the art on this test. We believe that the advantages that GEN3D provides over other models will enable it to contribute substantially to structure-based drug discovery efforts.