We investigate baroclinic instability in flow conditions relevant to hot extrasolar planets. The instability is important for transporting and mixing heat, as well as for influencing large-scale variability on the planets. Both linear normal mode analysis and non-linear initial-value calculations are carried out – focusing on the freely-evolving, adiabatic situation. Using a high-resolution general circulation model (GCM) which solves the traditional primitive equations, we show that large-scale jets similar to those observed in current GCM simulations of hot extrasolar giant planets are likely to be baroclinically unstable, on a timescale of a few to a few tens of planetary rotations, generating cyclones and anticyclones that drive weather systems. The growth rate and scale of the most unstable mode obtained in the linear analysis are in qualitative, good agreement with the full non-linear calculations. In general, unstable jets evolve differently depending on their signs (eastward or westward), due to the change in sign of the jet curvature. For jets located at or near the equator, instability is strong at the flanks – but not at the core. Crucially, the instability is either poorly or not at all captured in simulations with low resolution and/or high artificial viscosity. Hence, the instability has not been observed or emphasized in past circulation studies of hot extrasolar planets.