Abstract: Quantum simulation proposes to use future quantum computers to calculate properties of quantum systems. In the context of chemistry, the target is the electronic structure problem: determination of the electronic energy given the nuclear coordinates of a molecule. Since 2006 we have been studying quantum approaches to quantum chemical problems, and such approaches must face the challenges of high, but fixed, precision requirements, and fermion antisymmetry. I will describe several algorithmic developments in this area including improvements upon the Jordan Wigner transformation, alternatives to phase estimation, adiabatic quantum computing approaches to the electronic structure problem, methods based on sparse Hamiltonian simulation techniques and the potential for experiments realizing these algorithms in the near future.