One of the best known problem that a quantum computer is expected to solve more efficiently than a classical one is the simulation of quantum systems. While significant work has considered the case of discrete, finite dimensional quantum systems, the study of fast quantum simulation methods for continuous-variable systems has only received little attention. In this talk, I will present quantum methods to simulate the time evolution of two quantum systems, namely the quantum harmonic oscillator and the quantum particle in a quartic potential. Our methods are based on well-known product formulas for approximating the evolution and result in superpolynomial and polynomial quantum speedups, respectively. I will also present efficient quantum algorithms to prepare the eigenstates of the quantum harmonic oscillator that can be used to compute spectral properties and may be of independent interest. Generalizations and connections between our results and the so-called fractional Fourier transform, which is a generalization of the Fourier transform used in signal analysis, will also be discussed.