In this chapter, the authors delve a little deeper into the meaning of an asymptotic analysis of a phase field model. They guide the discussion by using the backdrop of the single-component and single-order parameter limit. Matched asymptotic boundary layer analysis is the formal name of the mathematical analysis used to map the behaviour of a set of phase field equations to a corresponding sharp interface model (SIM). The claim that a phase field model is quantitative simply means that model parameters and interpolation functions can be chosen such that its kinetics match those of a SIM in the limit of a diffuse interface width. Thin-interface analyses can converge onto SIM results for a range interface widths, making it possible to conveniently increase the interface width and decrease numerical resolution.