An overview of two categories of numerical approaches to phase change problems, front tracking and fixed domain methods, is given. A further comparison of several fixed domain methods is made with some benchmark test problems. The approximation of the latent heat for isothermal phase change is a key point for these numerical approaches. Generally the space-average approach is better than the others. The Euler backward, Crank-Nicolson, and Dupont schemes are the best for thermal analysis involving phase changes. The new effective and apparent capacity methods can eliminate the need of an artificially assumed temperature range for isothermal transformation and obtain accurate results. They are also computationally efficient compared with other methods. The new apparent capacity technique is invoked to model the isothermal part of phase change for several solidification models. The same approach is applied to the thermal analyses of the casting processes and the results are in good agreement with experiment data. A convective type interface element formulation is derived for the heat transfer at the interface of the cast and the mold, which can simplify the element matrix calculation. A thermal elasto-plastic stress model is derived and the public domain software NIKE2D is used for the stress analyses of the casting processes. The initial stage of ingot cooling in the mold and the continuous casting process are simulated. The latent heat released during solidification can affect the temperature field and the phase transformation in the solid state has an essential effect on the stress generation. The cast size, material hardening effect, and cast speed are all important factors for the temperature and stress distributions.