Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous diffusions and differential equations of fractional order. I will then present some recent results in the study of these two topics, including the counterpart of DeGiorgi-Nash-Moser-Aronson theory for non-local operators of fractional order. No prior knowledge in these two subjects is assumed.