Fourth-order partial differential equations arise in a wide range of applications, from thin plate and shell mechanics to phase-field models, fluid dynamics, and biological membranes. Their numerical approximation poses significant challenges, particularly when trying to avoid the construction of C1 conforming finite element spaces, which can be complex and restrictive. Nonstandard finite element methods - such as discontinuous Galerkin, C0 interior penalty, mixed formulations, virtual element methods, and Hybrid High Order methods - offer flexible alternatives that accommodate general meshes, local adaptivity, and robust handling of singularities and interfaces. In this minisymposium, we aim to discuss recent advances in the design, analysis, and efficient implementation of these nonstandard discretizations for both fourth-order problems and their extension to problems that require (local) H2-regularity. Topics include, but are not limited to, novel a priori and a posteriori error analysis, hp-adaptivity, structure-preserving schemes and the application of such methods to emerging problems in engineering and the sciences.