Self organised criticality : theory models and characterisation Giving a detailed overview of the subject this book takes in the results and methods that have arisen since the term self organised criticality was coined twenty years ago. Providing an overview of numerical and analytical methods from their theoretical foundation to the actual application and implementation the book is an easy access point to important results and sophisticated methods. Starting with the famous Bak Tang Wiesenfeld sandpile ten key models are carefully defined together with their results and applications. Comprehensive tables of numerical results are collected in one volume for the first time making the information readily accessible to readers. Written for graduate students and practising researchers in a range of disciplines from physics and mathematics to biology sociology finance medicine and engineering the book gives a practical hands on approach throughout. Methods and results are applied in ways that will relate to the readers own research When Bak Tang and Wiesenfeld (1987) coined the term Self Organised Criticality (SOC) it was an explanation for an unexpected observation of scale invariance and at the same time a programme of further research. Over the years it developed into a subject area which is concerned mostly with the analysis of computer models that display a form of generic scale invariance. The primacy of the computer model is manifest in the first publication and throughout the history of SOC which evolved with and revolved around such computer models. That has led to a plethora of computer models many of which are not intended to model much except themselves (also Gisiger 2001) in the hope that they display a certain aspect of SOC in a particularly clear way. The question whether SOC exists is empty if SOC is merely the title for a certain class of computer models. In the following the term SOC will therefore be used in its original meaning (Bak et al 1987) to be assigned to systems with spatial degrees of freedom [which] naturally evolve into a self organized critical point. Such behaviour is to be juxtaposed to the traditional notion of a phase transition which is the singular critical point in a phase diagram where a system experiences a breakdown of symmetry and long range spatial and in non equilibrium also temporal correlations generally summarised as (power law) scaling (Widom 1965a b; Stanley 1971)