The art of simulating quantum dynamics in recent years has become synonymous with Hamiltonian simulation. That is to say, we often assume implicitly that the quantum dynamical system of interest is specified by a sparse Hamiltonian matrix whose matrix elements are efficiently computable and use an oracle to input the Hamiltonian to a simulation algorithm. Hamiltonians, however, are not always available for simulating physical systems. This fact is particularly noticeable for simulations of quantum field theories wherein the Lagrangians are often easy to derive are often used rather than the Hamiltonian to simulate quantum dynamics using the path integral formalism. We address these problems in this paper by showing that quantum dynamics can be simulated efficiently using the path integral formalism using the Lagrangian in place of the Hamiltonian. The nature of the oracle used here is fundamentally different than that used in Hamiltonian simulation because the Lagrangian is a scalar and the Hamiltonian is a matrix. This means that our approach to simulation is fundamentally different than existing approaches and further allows quantum simulation to be performed for quantum field theories where the Hamiltonian is not explicitly known but the Lagrangian is simple to compute. We further investigate new simulation methods based on path integrals for Hamiltonians and show that in certain cases, such as near-adiabatic quantum dynamics, these simulation methods may have substantial advantages compared to conventional strategies.