Forward scattering solutions, whereby one predicts the scattered electromagnetic field given the goemetry, find numerous engineering applications in computer aided design. Therefore, a fast way to solve the forward scattering problem will impact a number of areas, like high- speed circuits, integrated optics, antenna analysis, remote sensing, geophysical sensing, and inverse scattering. We will describe several fast methods developed in our group to solve the forward scattering problem rapidly. These methods involve solving the volume integral equation of scattering as well as surface integral equation of scattering. Different strategies are used to accelerate the solutions of these integral equations. Both iterative and direct solution techniques will be considered. The computational complexity and memory requirement of various scattering algorithms will be discussed. In inverse scattering, one reconstructs the physical goemetry of a scatterer from the measured scattered field. Hence, it finds applications in image and profile reconstructions. When using a linear method, multiple scattering within a scatterer is ignored. By using a nonlinear inverse scattering method, such multiple scattering effect is accounted for. Image and profile reconstruction using such nonlinear inverse algorithm can remove artifacts that linear methods would not remove. We will discuss the use of the distorted Born iterative method and local shape function method to reconstruct a scattering object.