The field of partial differential equations (PDEs) is witnessing significant advancements in shape reconstruction and inverse problems. Researchers are developing innovative methods to reconstruct unknown shapes and sources from partial measurements, leveraging techniques such as monotonicity-based regularization, H(curl)-reconstruction, and Carleman estimates. These approaches are enabling the solution of complex inverse problems with improved accuracy and stability. Noteworthy papers include: A Monotonicity-Based Regularization Approach to Shape Reconstruction for the Helmholtz Equation, which proposes a novel convex data-fitting formulation for shape reconstruction. Conditional Stability and Numerical Reconstruction of a Parabolic Inverse Source Problem Using Carleman Estimates, which establishes conditional Lipschitz stability and proposes a numerical approach for solving a parabolic inverse source problem.