The Hawking mass is one of the important notions of mass in general relativity. We study the L²-gradient flow of the Hawking mass functional, which we will call the Hawking flow for short, on a closed surface in an asymptotically flat Riemannian 3-manifold. A highly related functional is the Willmore energy functional. Thus to begin with, we will first review the results developed for the L²-gradient flow of the Willmore energy functional, which is commonly known as the Willmore flow, on a closed surface in R³. We will sketch the proof for the longtime existence of the solution to the Willmore flow, with the main focus on developing the higher order estimates. Then, we shall discuss the difference between the two flows and the modifications required to establish the longtime existence for the Hawking flow.