The Chair of Quantitative Finance was created at École Centrale Paris, now CentraleSupélec, in 2007. Since its inception, most of its research activities were devoted to the study of high frequency financial data. The interdisciplinary nature of the team, composed of mathematicians, financial engineers, computer scientists and physicists, gave it a special dimension. A sizeable portion of its research efforts has been focused on the characterization and mathematical modelling of limit order books.

Literally at the core of every modern, electronic financial market, the limit order book has triggered a huge amount of research in the past twenty years, marked by the seminal work of Biais et al. (1995) on the empirical analysis of the Paris exchange and revitalized a few years later, in a fascinating manner, by the work of Smith et al. (2003). However, much as this topic is interesting, important and challenging, we realized that there was still no reference book on the subject! We therefore decided to assemble in a single document a survey of the existing literature and our own contributions on limit order books, whether they were pertaining to their statistical properties, mathematical modelling or numerical simulation.

We have tried to follow the intellectual approach of an experimental physicist: empirical data should come first, and only empirical analyses may be considered as a reliable ground for building up any kind of theory. The mathematical modelling follows. Models address the different phenomena that are observed and highlighted, and provide a framework to explain and reproduce these phenomena, and they are studied from theoretical, analytical and numerical perspectives.

The book is thus organized as follows: The first part is devoted to the empirical properties of limit order books; the second part, to their mathematical modelling and the third, to their numerical analysis. The fourth part deals with some advanced topics such as imperfection and predictability. Each part presents a survey of the existing scientific literature, as well as our own contributions.

Significant parts of the material covered in this book have already been presented in bits and pieces in different research and survey articles, in particular Chakraborti et al. (2011a,b); Abergel and Jedidi (2013, 2015); Anane and Abergel (2015); Muni Toke (2015, 2011).