Indoleamine 2,3-dioxygenase 1 (IDO1) catalyzes the rate-limiting step in the kynurenine pathway of tryptophan metabolism, which is involved in immunity, neuronal function, and aging. Its implication in pathologies such as cancer and neurodegenerative diseases has stimulated the development of IDO1 inhibitors. However, negative phase III clinical trial results of the IDO1 inhibitor epacadostat in cancer immunotherapy call for a better understanding of the role and the mechanisms of IDO1 inhibition. In this work, we investigate the molecular inhibition mechanisms of four known IDO1 inhibitors and of two quinones in detail, using different experimental and computational approaches. We also determine for the first time the X-ray structure of the highly efficient 1,2,3-triazole inhibitor MMG-0358. Based on our results and on a comprehensive literature overview, we propose a classification scheme for IDO1 inhibitors according to their inhibition mechanism, which will be useful for further developments in the field.