Preface; 1. Fundamental approximations; 2. Properties and derivatives; 3. Multivariate densities; 4. Conditional densities and distribution functions; 5. Exponential families and tilted distributions; 6. Further exponential family examples and theory; 7. Probability computation with p*; 8. Probabilities with r*-type approximations; 9. Nuisance parameters; 10. Sequential saddlepoint applications; 11. Applications to multivariate testing; 12. Ratios and roots of estimating equations; 13. First passage and time to event distributions; 14. Bootstrapping in the transform domain; 15. Bayesian applications; 16. Non-normal bases; References; Index.