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.