The motion of a quantum particle in a random magnetic flux in two dimensions is investigated. Two situations are distinguished, a ``Debye'' phase where the fluxes are uncorrelated, and a ``Meissner'' phase where the fluxes appear as neutral pairs. A geometrical interpretation of effective single-particle action in these phases is emphasized. Results are discussed for (a) a continuum white-noise model where we employ a trial-action method, (b) a continuum model with randomly distributed flux tubes where we obtain the form of the Lifschitz tail, and (c) a lattice model, where numerical results for the density of states and diamagnetic response of Debye and Meissner phases are given. An important conclusion is that the density of states in the Debye phase exhibits a sharp peak at an effective band edge.