The problem of biased diffusion in disordered media (percolation clusters) is analysed by means of the exit-time equation. Numerical simulations show that for percolation lattices tending to criticality, the volume-averaged exit time as a function of the Peclet number, Pe, deviates from the regular 1/Pe-behaviour and for high Pe grows monotonically with Pe. Numerical simulations on DLA-clusters and deterministic fractals indicate the applicability of the exit-time approach to singular fractal structures. Finally, exit-time analysis is adopted in explaining standard and non-standard features of dispersion of solute particles in highly heterogeneous porous packings.