We study radiatively cooling self-gravitating flows. The early transitional stage of cooling flows, from a static medium to a steady state inflow, as well as the implications for galaxy formation, are considered with this analysis. We find a self-similar solution to describe this inflow, and perform numerical modeling to test the breadth of initial conditions which will follow the analytical solutions. The self-similar solution incorporates a power law cooling term which is $\propto \rho^2 T^\lambda$; if $\lambda<1$ the central temperature increases with time. The paradoxical result of the temperature of cooling gas increasing with time is due to strong gravitational compression. This is confirmed with numerical simulations for $\lambda \leq -0.5$ that have moderately long central cooling times. Shorter cooling times lead to a central thermal runaway. For simulations with $\lambda > -0.5$, the center always cooled catastrophically and never followed the self-similar solution. We consider the observational consequences of the large luminosities calculated from these models.