We investigate the centrifugal instability of a pair of radially stratified immiscible liquids in the annular gap between concentric, corotating cylinders: two-fluid Taylor–Couette flow. Experiments show that a two-layer flow with a well-defined interface and Taylor vortices in each phase can be obtained. The experimental results are in good agreement with predictions of inviscid arguments based on a two-phase extension of Rayleigh’s criterion, as well as with detailed linear stability calculations. For a given geometry, the most stable configuration occurs for fluids of roughly (exactly in the inviscid limit) equal dynamic viscosities. For fluids with sufficiently low viscosity, we also experimentally observe another instability that is evidently the two-fluid analog of the instability of a thin film of liquid coating the inside of a rotating drum. ©1998 American Institute of Physics.