For a d-dimensional diffusion of the form dXt = μ(Xt)dt + σ(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second-order backward stochastic differential equation (2BSDE) $$dY_t = f(t,X_t, Y_t, Z_t, \Gamma_t) dt + Z_t'\circ dX_t, \quad t ın [0,T),$$ $$dZ_t = A_t dt + \Gamma_tdX_t, \quad t ın [0,T),$$ $$Y_T = g(X_T).$$ If the associated PDE $$- v_t(t,x) + f(t,x,v(t,x), Dv(t,x), D^2v(t,x)) = 0,$$ $$(t,x) ın [0,T) \times \cal R^d, \quad v(T,x) = g(x),$$ has a sufficiently regular solution, then it follows directly from Itô's formula that the processes $$v(t,X_t), Dv(t,X_t), D^2v(t,X_t), \cal L Dv(t,X_t), \quad t ın [0,T],$$ solve the 2BSDE, where 𝓁 is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Yt = v(t, Xt), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.