The recent developments in the research area have significantly advanced the field, particularly in the areas of numerical methods for partial differential equations (PDEs) on surfaces and innovative finite element techniques. A notable trend is the focus on high-order accuracy and efficiency in solving complex geometric problems, with several papers introducing novel methodologies that enhance computational capabilities. For instance, the integration of meshfree methods with Radial Basis Function-Finite Difference (RBF-FD) approximations has shown promise in handling PDEs on evolving surfaces, offering high-order convergence and flexibility. Additionally, the proposal of new interior penalty methods for higher-order formulations of surface Stokes problems has addressed the challenges associated with surface curvature, leading to positive definite discretizations that do not explicitly depend on Gauss curvature. These advancements are complemented by the development of optimization-based strategies for coupling 3D-1D problems, which have been extended to nonlinear contexts and applied with virtual element discretizations, enhancing the method's ability to manage geometric complexities. Furthermore, the introduction of predictor-corrector time-stepping methods in parametric finite element approaches for surface diffusion has achieved second-order temporal accuracy without the need for mesh regularization, maintaining long-term mesh equidistribution. These innovations collectively push the boundaries of what is computationally feasible, offering new tools for researchers in the field.

Noteworthy papers include one that presents a novel meshfree RBF-FD method for solving PDEs on surfaces, demonstrating high-order convergence and flexibility, and another that introduces a $C^0$ interior penalty method for the stream function formulation of the surface Stokes problem, which does not explicitly depend on the Gauss curvature of the surface.