Deep learning has been very successful in dealing with big data from various fields of science and engineering. It has brought breakthroughs using various deep neural network architectures and structures according to different learning tasks. An important family of deep neural networks are deep convolutional neural networks. We give a survey for deep convolutional neural networks induced by 1‐D or 2‐D convolutions. We demonstrate how these networks are derived from convolutional structures, and how they can be used to approximate functions efficiently. In particular, we illustrate with explicit rates of approximation that in general deep convolutional neural networks perform at least as well as fully connected shallow networks, and they can outperform fully connected shallow networks in approximating radial functions when the dimension of data is large.