Since A. P. Calderon's ground-breaking paper in 1980, inverse boundary value problems have received ever growing attention because of broad industrial, medical, and military applications. Exciting theorems have been proved about the uniqueness, stability, and range of the inverse problems. However, numerical solution of the inverse problems continues to be challenging since the problems are nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. In this talk, I report on progress of our research group over the past several years in mathematical analysis and computational studies of the inverse boundary value problems for the Helmholtz and Maxwell equations. I will present a continuation approach based on the uncertainty principle. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse medium scattering problems. I will also discuss convergence issues for the continuation algorithm and highlight ongoing projects in limited aperture imaging, and breast cancer imaging (dispersive medium).