In this work, we present a study on the theory of differential geometry of surfaces with a language of moving references. At first, we will introduce some groups of matrices and their actions in Euclidean space. Once this is done, we will talk about Euclidean mobile frames and show the procedure to obtain "the best possible frame". We use this to demonstrate Bonnet's existence and congruence theorems, and to find curvatures of some families of surfaces. Finally, we will introduce the general notion of Lie groups, Lie algebras and Lie group actions on differentiable manifolds, concluding that, through a generalization of the moving frame, Euclidean theory extends to any differentiable manifold, in particular, to the spherical and hyperbolic spaces.