Describes fifteen years' work which has led to the construc- tion of solutions to non-linear relativistic local field e- quations in 2 and 3 space-time dimensions. Gives proof of the existence theorem in 2 dimensions and describes many properties of the solutions.

I An Introduction to Modern Physics.- 1 Quantum Theory.- 2 Classical Statistical Mechanics.- 3 The Feynman-Kac Formula.- 4 Correlation Inequalities and the Lee-Yang Theorem.- 5 Phase Transitions and Critical Points.- 6 Field Theory.- Appendix to Part I. Hilbert Space Operators and Functional Integrals.- II Function Space Integrals.- 7 Covariance Operator = Green’s Function = Resolvent Kernel = Euclidean Propagator = Fundamental Solution.- 8 Quantization = Integration over Function Space.- 9 Calculus and Renormalization on Function Space.- 10 Estimates Independent of Dimension.- 11 Fields Without Cutoffs.- 12 Regularity and Axioms.- III The Physics of Quantum Fields.- 13 Scattering Theory: Time-Dependent Methods.- 14 Scattering Theory: Time-Independent Methods.- 15 The Magnetic Moment of the Electron.- 16 Phase Transitions.- 17 The ?4 Critical Point.- 18 The Cluster Expansion.- 19 From Path Integrals to Quantum Mechanics.- 20 The Polymer Expansion.- 21 Random Path Representations.- 22 Constructive Gauge Theory and Phase Cell Localization.- 23 Further Directions.

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