In the past few years there have been a number of major advances for quantum simulation.  I will be reviewing a number of them in this talk including the interaction picture simulation method, the QDRIFT algorithm, well conditioned multi-product formulas and new commutator bounds for high-order Trotter Suzuki methods which show that those algorithms are much more efficient than previously thought.  Finally I will discuss applications of these algorithms to problems in chemistry and simulation of the Schwinger model in 1+1D.