The primary instabilities and bicritical curves for flow between counter‐rotating cylinders have been computed numerically from the Navier–Stokes equations assuming axial periodicity. The computations provide values of the Reynolds numbers, wavenumbers, and wave speeds at the primary transition from Couette flow for radius ratios from 0.40–0.98. Particular attention has been focused on the bicritical curves that separate (as the magnitude of counter‐rotation is increased) the transitions from Couette flow to flows with different azimuthal wavenumbersmandm+1. This lays the foundation for further analysis of nonlinear mode interactions and pattern formation occurring along the bicritical curves and serves as a benchmark for experimental studies. Preliminary experimental measurements of transition Reynolds numbers and wave speeds presented here agree well with the computations from the mathematical model.