Many linear and nonlinear partial differential equations (PDEs) arise from the minimization of an underlying energy functional. Examples are ubiquitous and arise, for instance, in the study of non-Newtonian fluids, minimal surfaces, or nonlinear mechanics. Whereas classical numerical methods and their analysis mostly focus entirely on the solution of the associated PDE/Euler-Lagrange equations, recent contributions have started to take the energetic structure into account. This is particularly relevant for engineering applications where the energetic behavior of minimizers can be more relevant than the solution itself. In addition, analysis in terms of energy often naturally fits the structure of the problem, which can be exploited to derive error bounds with explicit constants. We will introduce the method of flux-equilibration which yields highly effective a posteriori error bounds that can subsequently be used to steer adaptive mesh-refinement. After a general introduction to the concept, we highlight recent advances and generalizations to the nonlinear setting and present numerical results highlighting the efficiency of the method.