Results of a spatial stability analysis and of numerical simulations of a “slab” jet in which optically thin radiative cooling is dynamically important are presented. Two different cooling curves are used. Unstable Kelvin-Helmholtz modes are significantly different from the adiabatic limit, and the form of the cooling function strongly affects the results. The numerical simulations are in excellent agreement with the linear stability analysis. In the non-linear regime growth of the surface wave at low frequencies results in sinusoidal oscillation which can disrupt the jet, while non-linear body waves produce low amplitude wiggles within the jet that can result in shocks within the jet. In cooling jets, these shocks can produce dense knots and filaments of cooling gas within the jet, and weak shock “spurs” in the ambient gas. Acceleration of ambient gas can be produced by these “spurs”, or by rapid entrainment if the jet is disrupted. For parameters typical of protostellar jets perturbations with a period of < 100 yrs should excite body waves which produce internal shocks and small amplitude wiggles. The lack of large amplitude wiggles in most observed systems is consistent with the suggestion that jets arise from the inner regions (r < 1 AU) of accretion disks.