Two of the fundamental issues in the analysis and applications of nonlinear partial differential equations in Science and Engineering are weak rigidity/continuity and compactness/convergence. These issues involve rigidity problems and singular limit problems in geometry and mechanics, convergence problems in numerical analysis, and hydrodynamics limits in statistical/continuum physics, among others. In this talk, we will discuss some recent developments in the analysis of several longstanding problems involving weak rigidity and compactness for fundamental nonlinear partial differential equations in geometry, mechanics, and other areas. In particular, these problems include the inviscid limit of the compressible Navier-Stokes equations to the Euler equations, the construction of global entropy solutions of spherically symmetric solutions of the multidimensional compressible Euler equations, the construction of stochastic entropy solutions of nonlinear hyperbolic conservation laws with random forcing, the sonic-subsonic limit of approximate or exact solutions of the multidimensional steady Euler equations, and the weak rigidity of isometric embeddings on Riemannian, or even semi-Riemannian, manifolds (weak continuity of the Gauss-Codazzi-Ricci equations on the manifolds). Further trends, perspectives, and open problems will also be addressed.