On-line learning is one of the most commonly used techniques for training neural networks. Though it has been used successfully in many real-world applications, most training methods are based on heuristic observations. The lack of theoretical support damages the credibility as well as the efficiency of neural networks training, making it hard to choose reliable or optimal methods. This book presents a coherent picture of the state of the art in the theoretical analysis of on-line learning. An introduction relates the subject to other developments in neural networks and explains the overall picture. Surveys by leading experts in the field combine new and established material and enable nonexperts to learn more about the techniques and methods used. This book, the first in the area, provides a comprehensive view of the subject and will be welcomed by mathematicians, scientists and engineers, both in industry and academia.