We study the mixing time of some Markov Chains converging to critical physical models. These models are indexed by a parameter \beta and there exists some critical value \beta c where the model undergoes a phase transition. According to Physics lore, the mixing time of such Markov Chains is often of logarithmic order outside the critical regime, when \beta \ne \beta c, and satisfies some power law at criticality, when \beta = \beta c. We prove this in the two following settings: 1. Lazy random walk on the critical percolation cluster of "mean-field" graphs, which include the complete graph and random d-regular graphs. The critical mixing time here is of order \Theta (n). This answers a question of Benjamini, Kozma and Wormald [4]. 2. Swendsen-Wang dynamics [33] on the complete graph. The critical mixing time here is of order \Theta (n^{1/4} ). This improves results of Cooper, Dyer, Frieze and Rue [9]. In both settings, the main tool is understanding the Markov Chain dynamics via properties of critical percolation on the underlying graph.