Preface -- 1. Introduction -- 1.1. Research into and interest in the long air gap discharge -- 1.2. Scope and objectives -- 1.3. Intended audience2. The background of air gap discharge theory -- 2.1. Introduction -- 2.2. Ionization phenomena -- 2.3. Cross section and mean free path--elastic collisions -- 2.4. Mobility, diffusion, and recombination -- 2.5. Discharge in small air gaps : Townsend's discharge theory -- 2.6. Self-sustaining discharge -- 2.7. Limits of Townsend's theory -- 2.8. Streamer-leader theory3. The positive discharge in long air gaps -- 3.1. Introduction -- 3.2. Air gap breakdown process under an impulse voltage4. The negative discharge in long air gaps -- 4.1. Introduction -- 4.2. First negative corona and stem -- 4.3. The cathodic stem and upward discharges -- 4.4. The negative leader -- 4.5. The space stem or pilot system -- 4.6. Final jump phase5. Lightning discharge -- 5.1. Introduction -- 5.2. The global electric circuit -- 5.3. The most common types of lightning discharge -- 5.4. A description of a cloud-ground lightning discharge processes -- 5.5. Lightning electrical parameters -- 5.6. Comparison of laboratory sparks and cloud-ground lightning discharges6. A review of existing mathematical models developed for long air gap discharges -- 6.1. Introduction -- 6.2. Positive discharge models -- 6.3. Negative discharge models -- 6.4. Fractal models of long discharges7. Modelling the positive discharge in long air gaps -- 7.1. Introduction -- 7.2. A general description of the dynamic procedure -- 7.3. The applied voltage wave shape -- 7.4. The characterization of the discharge propagation -- 7.5. Distributed-circuit-based modelling -- 7.6. The distributed-circuit elements -- 7.7. General flowchart of the model -- 7.8. Extension to a very long air gap : positive lightning8. Modelling the negative discharge in long air gaps -- 8.1. Introduction -- 8.2. The development of a negative discharge -- 8.3. Theoretical background -- 8.4. Distributed-circuit-based modelling -- 8.5. General description of computation steps -- 8.6. Extension to a very long air gap : negative lightning9. Applications of the model developed for positive discharge in long air gaps -- 9.1. Introduction -- 9.2. Prediction of the characteristics of long air gap discharges : simulations of some laboratory experiments -- 9.3. Prediction of the switching impulse withstand voltages of long air gaps -- 9.4. Flashover voltage of long air gaps in the presence of a floating insulating barrier10. Applications of the model developed for negative discharge in long air gaps -- 10.1. Introduction -- 10.2. Simulation of laboratory experiments -- 10.3. Prediction of the 50% negative breakdown voltage11. Application of the model to positive lightning discharge -- 11.1. Introduction -- 11.2. Prediction of positive lightning discharge parameters -- 11.3. Influence of soil conductivity and cloud-ground distance on the positive lightning impulse current -- 11.4. Electric field changes of the leader and return stroke -- 11.5. Magnetic field associated with the leader12. Application of the model to the process of lightning-ground connection and quantification of the striking distance -- 12.1. Introduction -- 12.2. Modelling the lightning connection process to a ground structure -- 12.3. A quantitative study of lightning striking distance factors13. Application of the model to evaluate the induced effects on overhead lines due to a nearby positive lightning downward leader -- 13.1. Introduction -- 13.2. Induced effects on an overhead line due to nearby positive lightning downward leader14. Negative lightning model--applications -- 14.1. Introduction -- 14.2. The prediction of negative lightning discharge parameters -- 14.3. Electric and magnetic fields associated with the leader.