Quantum computing has rapidly advanced in recent years due to substantial development in both hardware and algorithms. These advances are carrying quantum computers closer to their impending commercial utility. Drug discovery is a promising area of application that will find a number of uses for these new machines. As a prominent example, quantum simulation will enable faster and more accurate characterizations of molecular systems than existing quantum chemistry methods. Furthermore, algorithmic developments in quantum machine learning offer interesting alternatives to classical machine learning techniques, which may also be useful for the biochemical efforts involved in early phases of drug discovery. Meanwhile, quantum hardware is scaling up rapidly into a regime where an exact simulation is difficult even using the world’s largest supercomputers. We review how these recent advances can shift the paradigm with which one thinks about drug discovery, focusing on both the promises and caveats associated with each development. In particular, we highlight how hybrid quantum-classical approaches to quantum simulation and quantum machine learning could yield substantial progress using noisy-intermediate scale quantum devices, whereas fault-tolerant, error-corrected quantum computers are still in their development phase.