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The present work is divided into two parts. The first one is focused on the study of the seminal works which are related with the origins of the diffusion concept in physics, for instance, the works of Fourier, Einstein, Brown, Rayleigh, Fick, among others. In the second part we studied the origins and the definitions of the anomalous diffusion. We also showed some mathematical approaches to obtain the anomalous diffusive behavior. Finally, we investigate solutions, by using the Green function approach, for a system governed by a non-Markovian Fokker-Planck equation that are related to the comb model. For this system, we consider an arbitrary initial condition, in the presence of time dependent diffusion coeffcients and spatial fractional derivative, and analyze the connection to the anomalous diffusion.