Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoreti- cal foundation for understanding the approximation or generalization ability of deep learning methods with network architectures such as deep convolutional neural networks with convo- lutional structures. This talk describes a mathematical theory of deep convolutional neural networks (CNNs). In particular, we show the universality of a deep CNN, meaning that it can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. Our quantitative estimate, given tightly in terms of the number of free parameters to be computed, verifies the efficiency of deep CNNs in dealing with large dimensional data. Some related distributed learning algorithms will also be discussed.