Abstract: Numerical methods for partial differential equations (PDEs) are the main research area in computational mathematics. It includes finite difference methods, finite element methods, boundary element methods and spectral methods. In this paper, some contemporary topic and advanced development on numerical methods for PDEs will be introduced. The main focuses are on some applied science or engineering areas such as optimal control problem governed by partial differential equations, miscible displacement problems of one incompressible fluid by another in a porous medium, nonlinear reaction-diffusion equations, convection-dominated convection-diffusion problems, Volterra integral and differential equations. The basis of these new numerical techniques lies in the application of high accuracy, post-processing, super-convergence, two-grid methods, adaptive mesh refinement, adaptive moving mesh methods, and spectral methods.