When physicists became convinced that matter was not continuous but made from atoms, new ideas on old subjects started flourishing. Not only well-known macroscopic laws (thermodynamics, hydrodynamics) became better understood and bolstered by a more fundamental underlying reality, but a host of spectacular and often unexpected effects were rationalized, in particular collective emergent phenomena phase transitions, superconductivity, avalanches, etc. Similarly, after decades of mathematical finance devoted to the study of effective low frequency models of markets (chiefly based on variations on the Brownian motion), the increasing availability of high frequency data now allows a comprehensive study of price formation and of the microstructure of supply and demand. A new era of financial modelling is opening up, with the hope of addressing a hitherto neglected yet crucial aspect of price dynamics: feedback effects that can lead to market anomalies, instabilities and crashes. Instead of considering the market as an inert, reliable measurement apparatus that merely reveals the fundamental value of assets without influencing it, the empirical study of the order book reveals that markets do generate their own dynamics. New intuitions about market dynamics are necessary. New fascinating statistical regularities are collected and modelled, in particular using numerical simulations of agent based models. New analytical tools are being built to account for these observations. The final goal is, much as in physics, to understand the emergent phenomena and replace ad-hoc models of prices by micro-founded ones where jumps, fat-tails and clustered volatility would have a clear origin. This is important on many counts: while the intellectual endeavour is of course exciting in itself, its offshoots will deeply influence the way we think about market regulation in the wake of high-frequency trading, and the models we use for financial engineering (from derivative pricing to algorithmic trading and optimal execution).

Limit order books offers a much needed, broad review of a field that has literally exploded in the last 20 years, where researchers from economics, financial mathematics, physics, computer science, etc. compete and confront. This diversity is well illustrated by the content of the present book that covers a very wide ground, from empirical facts to advanced mathematical techniques and numerical simulation tools. It will be a very useful and inspiring entry point for all scientists, engineers, regulators and traders interested in understanding how financial markets really work at the basic level.