In this talk, we present recent advances in the numerical analysis of solution-driven surface evolution, spanning geometric flows, two-phase flows, and fluid-structure interactions. We report a matrix-vector framework for deriving rigorous error estimates of finite element approximations to an evolving surface driven by the solutions of PDEs on the surface itself, as well as interfaces driven by fluids on both sides or coupled with elastic structures. Our analysis addresses the interplay between errors in computing interface location and PDE solutions, providing error bounds for the problem with unknown surface/interface evolution. The rigorous error analysis allows us to understand the stability of numerical algorithms and to design more stable algorithms to preserve high-quality meshes during computations, including artificial tangential motions on the evolving surfaces and interfaces, and artificial mesh velocity which improves the bulk meshing for fluid free-boundary problems. These methods enhance the accuracy and efficiency in simulating complex interfacial dynamics.