Quantum chemistry is expected to be one of the first areas where quantum computers can demonstrate an advantage over classical computers. Many algorithms are proposed for the key tasks in quantum chemistry, namely finding accurate molecular ground and excited states of a molecular Hamiltonian. At the same time, there are several challenges before these devices can assist research in chemical sciences. I will outline the research problem, recent developments in this exciting new field and some of the major theoretical challenges that need to be resolved in the future. Further, I will discuss our new developments to tackle these challenges, particularly, pulse-level variational molecular simulation algorithm (ctrl-VQE) and a new molecular excited-state method (q-sc-EOM) aimed at near-term quantum computing.