Title: A Priori Estimates  of the Generalization Error for Two-layer Neural Networks. Abstract: New estimates for the generalization error are established for a nonlinear regression problem using a two-layer neural network model. These new estimates are a priori in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted,  not the parameters in the model.  In contrast,  most existing results for neural networks are a posteriori in nature in the sense that the bounds depend on some norms of the model parameters. The error rates are comparable to that of the Monte Carlo method in terms of the size of the dataset.  Moreover, these bounds are equally effective in the over-parametrized regime when the network size is much larger than the size of the dataset. 