Finite Element Methods (FEM) are powerful tools for the numerical solution of partial differential equations (PDEs) in a wide range of applications. The versatility of FEM allows the development of problem-adapted numerical schemes tailored to the specific characteristics of the PDEs under consideration. In this presentation, we review several variants of FEM that have been tailored to address specific classes of PDE problems, ranging from those on moving domains to incompressible flows and stellar oscillations. We discuss the challenges associated with these problems, the development of suitable FEM discretisations and their analysis. A powerful subclass of FE methods is the class of Discontinuous Galerkin (DG) methods, also called non-conforming methods, which offer more flexibility compared to conforming FEM and can be used for different purposes, e.g. to deal with stability issues in the context of convection-dominated problems, or to bypass the construction of conforming finite element spaces. This flexibility typically comes at a higher computational cost. We introduce and compare two approaches to mitigate this cost: Hybrid DG (HDG) and Trefftz DG methods. We discuss the theoretical and implementation aspects of these methods and present numerical results illustrating their performance.