Multigrid algorithms for discretized nonlinear partial differential equations and systems are nearly as old as multigrid itself. Over the years several approaches and variants of nonlinear multigrid algorithms have been developed. Typically, for relatively easy problems the different approaches exhibit similar performance. However, for difficult problems the behavior varies, and it is not easy to predict which approach may prevail.   In this talk we will consider nonlinear multigrid, focusing on the task of coarse-grid correction, in a general framework of variational coarsening. Such a view reveals clear relations between the various existing approaches and may suggest future variants. This study also sheds light on the choice of inter-grid transfer operators, which are so important for obtaining fast multigrid convergence, and which have received much attention in linear multigrid algorithms but far less so in nonlinear multigrid.