During my study leave in 2018 I have applied nonlinear stability analysis techniques to the Douglas-Rachford Algorithm, with the aim of shedding light on the interesting non-convex case, where convergence is often observed but seldom proven. The Douglas-Rachford Algorithm can solve optimisation and feasibility problems, provably converges weakly to solutions in the convex case, and constitutes a practical heuristic in non-convex cases. Lyapunov functions are stability certificates for difference inclusions in nonlinear stability analysis. Some other recent nonlinear stability results are showcased as well.