In recent years, the image and video coding technologies have advanced by leaps and bounds. However, due to the popularization of image and video acquisition devices, the growth rate of image and video data is far beyond the improvement of the compression ratio. In particular, it has been widely recognized that there are increasing challenges of pursuing further coding performance improvement within the traditional hybrid coding framework. Deep convolution neural network (CNN) which makes the neural network resurge in recent years and has achieved great success in both artificial intelligent and signal processing fields, also provides a novel and promising solution for image and video compression. In this paper, we provide a systematic, comprehensive and up-to-date review of neural network based image and video compression techniques. The evolution and development of neural network based compression methodologies are introduced for images and video respectively. More specifically, the cutting-edge video coding techniques by leveraging deep learning and HEVC framework are presented and discussed, which promote the state-of-the-art video coding performance substantially. Moreover, the end-to-end image and video coding frameworks based on neural networks are also reviewed, revealing interesting explorations on next generation image and video coding frameworks/standards. The most significant research works on the image and video coding related topics using neural networks are highlighted, and future trends are also envisioned. In particular, the joint compression on semantic and visual information is tentatively explored to formulate high efficiency signal representation structure for both human vision and machine vision, which are the two dominant signal receptor in the age of artificial intelligence. Image and Video Compression with Neural Networks: A Review