The Generalized Elastic Model accounts for the dynamics of several physical systems, such as polymers, fluctuating interfaces, growing surfaces, membranes, proteins and file systems among others. We derive the fractional stochastic equation governing the motion of a probe particle (tracer) in such kind of systems. This Langevin equation involves the use of fractional derivative in time and satisfies the Fluctuation-Dissipation relation, it goes under the name of Fractional Langevin Equation. Within this framework the spatial correlations appearing in the Generalized Elastic Model are translated into time correlations described by the fractional derivative together with the space-time correlations of the fractional Gaussian noise. We derive the exact scaling analytical form of several physical observables such as structure factors, roughness and mean square displacement. Special attention will be devoted to the dependence on initial conditions and linear-response relations in the case of an applied potential.