The computational design of molecules and materials can help make progress in many of the most pressing world problems such as health, energy and climate. Many problems in these domains can benefit from a new generation of inexpensive, stable and abundant materials. National initiatives such as the Materials Genome Initiative have embraced high-throughput discovery approaches for accelerating the discovery cycles. A key component for the success of this vision and for making a dent in these problems is the availability of accurate, hopefully exact quantum calculations of the properties of candidate molecules and materials. I will begin this talk by briefly discovering the opportunities and challenges of molecular discovery by using an example of the development of molecular flow batteries from my laboratory. Quantum computers are able to perform exact (within a basis) simulation of chemical systems and materials. I will then discuss and briefly summarize the different approaches for carrying out quantum chemistry on a quantum computer. I will then focus on the variational quantum eigensolver (VQE). The VQE approach promises to be a powerful method for employing near-term quantum devices for carrying out quantum chemical simulations. I will share some ideas that we are currently working on and directions for the future.