The collection of all symmetries of rotation and reflection of a plane figure is used as an example to motivate a review of the concept of group, from its abstract definition to the consideration of other notions such as subgroups and conjugacy. A system consisting of several plane figures is then exploited to demonstrate that in addition to the individual symmetries of each of the figures (which may be called local symmetries) it may be worth considering also distant symmetries of congruence between two or more of these figures. The incorporation of both local and distant symmetries under a single mathematical umbrella leads to the notion of groupoid, whose precise definition in a general setting that transcends these examples is provided in detail and illustrated with other, not necessarily geometric, examples. A groupoid can be visualized as a set of arrows hovering over a set of objects, called the base of the groupoid.