Anomalous transport phenomena emerge in systems where quantum anomalies break certain classical symmetries leading to nonconservation of associated, and otherwise classically conserved, currents. In backgrounds of magnetic or gravitational fields, as well as in rotating systems, the anomalies may generate nondissipative flows of energy and charge currents even in the presence of strong interactions. We review recent theoretical developments on the anomalous transport in systems possessing axial, axial-gravitational, and conformal anomalies. We discuss the experimental consequences of these anomalies in recently discovered  Dirac and Weyl semimetals which are three-dimensional crystals whose low energy excitations are solutions of the massless Dirac equation. We concentrate on the consequences of gravitational and conformal anomalies for thermal and charge transport in curved spacetimes and electromagnetic backgrounds.