We present theoretical and computational descriptions of the dynamics of multicomponent lipid bilayer membranes. These systems are both model systems for "lipid rafts" in cell membranes and interesting physical examples of quasi-two-dimensional fluids. Our chief tool is a continuum simulation that uses a phase field to track the composition of the membrane, and solves the hydrodynamic equations exactly using the appropriate Green's function (Oseen tensor) for the membrane. We apply this simulation to describe the diffusion of domains in phase-separated membranes, the dynamics of domain flickering, and the process of phase separation in lipid membranes. We then derive an analytical theory to describe domain flickering that is consistent with our simulation results, and use this to analyze experimental measurements of membrane domains. Through this method, we measure the membrane viscosity solely from fluorescence microscopy measurements. We study phase separation in quasi-two-dimensional membranes in depth with both simulations and scaling theory, and classify the different scaling regimes and morphologies, which differ from pure two-dimensional fluids. Our results may explain previous inconsistent measurements of the dynamical scaling exponent for phase separation in membranes. We also extend our theory beyond the simplest model, including the possibility that the membrane will be viscoelastic, as well as considering the inertia of the membrane and the fluid surrounding the membrane.