Much of geometric group theory can be restated in more general terms using the language of coarse spaces and coarse geometry. This is a very natural language that has been largely overlooked: this talk aims to remedy this situation. I will give a gentle introduction to the notions of (metric) coarse groups and their coarse actions. Rather than going for hard results, I will put emphasis on concrete examples and connections with classical notions and results. This will exemplify the conceptual clarity of the categorical/coarse geometric approach and point to various natural questions.