This paper has expository nature and it is a survey on the meaning of symmetry for a topologist. The main purpouse is to give an approximation to the precise concept of equivariant fundamental grupoid starting from the one of ordinary fundamental grupoid. This concept is closely related to the one of equivariant covering space, so we start also dealing with ordinary covering spaces. For didactical reasons some classical results are presented, as well as some relatively new. A few proof are inserted when they are short enough and contribute with clarity.