Within the last several years there has been tremendous growth in quantum algorithms for Hamiltonian simulation which have led not only to advances for simulating the underlying dynamics in chemistry and condensed matter systems but has also led to new algorithms for solving linear systems, semidefinite programming and a host of other applications. In this talk, I will provide a high-level overview of the key strategies employed in modern Hamiltonian simulation algorithms. I will aim to not only show how modern quantum simulation algorithms work but also show how such algorithms can be applied to take advantage of different features of a problem such as commutativity or diagonal dominance of the Hamiltonian. I will then show how in practice these methods can be chosen to optimize simulations of chemistry and simulations of quantum electrodynamics in quantum systems. Finally, I will conclude by presenting open problems and new opportunities that these new paradigms create for developing new algorithms for simulation and beyond.