Anomalous diffusion and slow non-exponential relaxation in biomolecular systems are studied within the theoretical framework of non-equilibrium statistical physics. The focus is on the relation between transport coefficients and the asymptotic form of statistical observables, such as the mean square displacements of the diffusing particles and related time correlation functions within the framework of the Generalized Langevin Equation. It is shown that fractional diffusion equations describe anomalous diffusion asymptotically correctly. In this context, molecular dynamics simulations are an essential tool to illustrate theoretical results, in particular the relation between anomalous diffusion and the caging effect of diffusing particles in their respective environments. Examples are presented for the lateral diffusion of molecules in lipid bilayers and for the relaxation backbone dynamics in proteins.