This volume contains the proceedings of two AMS Special Sessions ``Recent Developments on Analysis and Computation for Inverse Problems for PDEs,'' virtually held on March 13-14, 2021, and ``Recent Advances in Inverse Problems for Partial Differential Equations,'' virtually held on October 23-24, 2021. The papers in this volume focus on new results on numerical methods for various inverse problems arising in electrical impedance tomography, inverse scattering in radar and optics problems, reconstruction of initial conditions, control of acoustic fields, and stock price forecasting. The authors studied iterative and non-iterative approaches such as optimization-based, globally convergent, sampling, and machine learning-based methods. The volume provides an interesting source on advances in computational inverse problems for partial differential equations.