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The structure and evolution of radiatively cooling, supersonic jets are explored by means of two-dimensional numerical simulations. The simulations reveal that the basic morphology of cooling jets is similar to that of adiabatic outflows, but that there are also several important differences. In particular, it is found that a dense, cold shell condenses out of the shocked gas at the bead of the jet provided that the cooling distance behind either one of the two principal shocks (the leading bow shock, which propagates into the ambient medium, and the jet shock, where the beam gas is thermalized) is smaller than the jet radius. The shell is dynamically unstable and eventually fragments into several distinct clumps. Radiative cooling also decreases the size and pressure of the "cocoon" of shocked jet material and of the surrounding "shroud" of shocked ambient gas, resulting in a reduced collimation of the supersonic beam and in fewer and weaker internal shocks. The decrease in collimation leads, in turn, to systematically lower propagation speeds for strongly cooling jets. However, as the cooling rate becomes even larger, the strong cooling behind internal jet shocks tends to enhance the collimation by reducing the thermal pressure in the beam. For very high cooling rates, the material that accumulates at the head of the jet forms an extended plug of cold gas that is reminiscent of the "nose cone" seen in numerical simulations of strongly magnetized, adiabatic jets. Conical internal shocks that are consistent with the reflection pinch modes of the Kelvin-Helmholtz instability are seen in some of the simulations and produce center-brightened knots of low-excitation emission whose densities are significantly higher than the initial jet density. We investigate the dependence of the jet properties on the density ratio between the beam and the ambient medium and on the strength of the radiative cooling. We analyze the relevant dynamical and thermal instabilities and discuss the implications of these results to the interpretation of stellar jets and Herbig-Haro objects.