We present some development and applications of the level set method in several fields of engineering, more specifically in multiphase flows and materials science. A ghost-fluid method is described for the simulation of multiphase flows with phase change and its application to thin film boiling. We also describe recent numerical advances for the simulation of moving boundary problems on non-graded adaptive Cartesian grids with application to the Navier-Stokes equations and the Stefan problem. Novel discretizations of the Heaviside and Dirac delta functions are presented. A hallmark of these discretizations is that they are robust to the perturbations of the interface's location on the grid. Results are also provided on a second-order accurate computation of interface curvature.