In 1979, Klainerman proved the first global existence result for fully nonlinear hyperbolic equations using Nash-Moser's method. The result was a breakthrough in the field. In the present talk, I shall report on the corresponding study for fully nonlinear stochastic hyperbolic equations driven by a Wiener process. The main result of the research is stochastic counterpart of Klainerman's global existence theorem. The main tool for our investigation is Nash-Moser's iteration scheme combined with stochastic calculus and the theory of enlargement of filtrations.