Elements of the classical field theory based on a variational formulation of the Hamilton type are discussed and corresponding 4- dimensional Lagrange formalism is presented both as the variational and the group theoretical script. Variational symmetries (geometric and generalized) of field equations and the Noether theorem providing a regular way of obtaining a conservation law for every given variational symmetry are revisited in the study in order to give a complete version of the contemporary field theory. All developments are presented in the non-linear frame (i.e.
