This special issue is dedicated to Professor Peter Monk in honor of his contribution and leadership in the areas of direct and inverse scattering theory and the numerical analysis of partial differential equations and integral equations, for more than 25 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Computational Mathematics and Inverse Problems honoring Peter Monk, held at Michigan Tech University, Houghton, Michigan, August 15–19, 2016. As organizers of this conference and close collaborators of Peter Monk, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a special occasion that gathered together long term colleagues, collaborators, former students and friends of Professor Monk. It gave us great pleasure to be guest editors of this special issue of the CAMWA that is a collection of original research papers on topics on numerical methods for PDEs and computational inverse problems. Much of the work presented here has been directly or indirectly influenced by the work of Peter Monk, offering the reader a glimpse of his significant impact in these research areas. We would like to thank all authors who have contributed a paper for this special issue. Special thanks goes to the Editor in Chief of Computers and Mathematics with Applications, Leszek Demkowicz, for supporting and facilitating this publication. We would also like to thank all the participants of the International Conference on Computational Mathematics and Inverse Problems who made such a successful, stimulating and pleasant event possible. Last, but not least, we would also like to thank the sponsors of the conference: Kliakhandler Fellowship, Michigan Tech, University of Delaware, and the National Science Foundation.