Title: Exploring Chemical Space with Score-based Out-of-distribution Generation. Abstract: A well-known limitation of existing works on molecule generation is that the generated molecules highly resemble those in the training set. To generate truly novel molecules with completely different structures that may have even better properties than known molecules for de novo drug discovery, more powerful exploration in the chemical space is necessary. To this end, we propose Molecular Out-Of-distribution Diffusion (MOOD), a novel score-based diffusion scheme that incorporates out-of-distribution (OOD) control in the generative stochastic differential equation (SDE) with simple control of a hyperparameter, thus requires no additional computational costs unlike existing methods (e.g., RL-based methods). However, some novel molecules may be chemically implausible, or may not meet the basic requirements of real-world drugs. Thus, MOOD performs conditional generation by utilizing the gradients from a property prediction network that guides the reverse-time diffusion process to high-scoring regions according to multiple target properties such as protein-ligand interactions, drug-likeness, and synthesizability. This allows MOOD to search for novel and meaningful molecules rather than generating unseen yet trivial ones. We experimentally validate that MOOD is able to explore the chemical space beyond the training distribution, generating molecules that outscore ones found with existing methods, and even the top 0.01% of the original training pool.