I will survey some fairly universal mechanisms which induce slow regimes for random walks. I will mainly give a few examples of dynamics in random media, where the complex geometry of the medium can induce interesting slow diffusion, in particular when a drift is added. The paradigm will be random walks on random trees or on percolation clusters. I will report on recent joint works with M.Cabezas, J. Cerny, A.Fribergh, A.Hammond, Y. Hu, N.Gantert, S. Olla, R. Royfman, V. Sidoravicius, and O.Zeitouni.