We review our construction of a Morse homology theory for the Yang-Mills gradient flow in two dimensions and its relation to Weber's heat flow homology. We discuss compactness and Morse-Smale transversality for the perturbed flow, which invokes a novel $L^2$ local slice theorem due to Mrowka-Wehrheim. Finally we show how a modified Yang-Mills functional leads to an "elliptic Yang-Mills flow" for which a Floer type homology theory is currently under construction. (Last part in joint work with R. Janner).