Explicit stabilized methods are an efficient and powerful alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimensions. In the present thesis, we derive new explicit stabilized methods for stiff and ergodic stochastic differential equations and for stiff optimal control problems. We analyze rigorously their stability and convergence properties. Numerical experiments, including deterministic and stochastic diffusion partial differential equations, illustrate the performance of the proposed schemes.