This review is an expository treatment of the displacement of one fluid by another in a two-dimensional geometry (a Hele-Shaw cell). The Saffman-Taylor equations modeling this system are discussed. They are simulated by random-walk techniques and studied by methods from complex analysis. The stability of the generated patterns (fingers) is studied by a WKB approximation and by complex analytic techniques. The primary conclusions reached are that (a) the fingers are linearly stable even at the highest velocities; (b) they are nonlinearly unstable against noise or an external perturbation; the critical amplitude for the noise being an exponential function of a power of the velocity for high velocities; (c) such exponentials seem to dominate high-velocity behavior; as can be seen from a WKB analysis; and (d) the results of the Saffman-Taylor equations disagree with experiments; apparently because they leave out film-flow phenomena.
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