This paper introduces a novel machine learning algorithm, termed the deep 2BSDE method, for solving high-dimensional fully nonlinear second-order partial differential equations (PDEs) and second-order backward stochastic differential equations (2BSDEs). Such PDEs are prevalent in quantitative finance, for example in derivative pricing under Knightian uncertainty, portfolio optimization, and credit valuation adjustment models, but their high dimensionality typically renders classical numerical methods infeasible due to the curse of dimensionality. The key innovation is to reformulate the PDE problem using a connection between fully nonlinear PDEs and 2BSDEs, leading to a merged formulation that can be discretized forward in time. This discretization involves unknown functions (the Hessian of the solution and a drift term) which are then approximated by deep neural networks. The parameters of these networks, along with the initial values of the solution and its gradient, are learned by minimizing a loss function that measures the error between the prescribed terminal condition and the forward simulation of the 2BSDE. The optimization is performed using stochastic gradient descent-type methods. The authors demonstrate the effectiveness of their approach through numerical experiments on several high-dimensional problems, including a 100-dimensional Black-Scholes-Barenblatt equation, a 100-dimensional Hamilton-Jacobi-Bellman equation, and nonlinear expectations of G-Brownian motion in 1 and 100 dimensions. The results show that the method can accurately approximate solutions in high dimensions, overcoming the curse of dimensionality and providing a practical tool for solving fully nonlinear PDEs arising in financial engineering and other fields.