I The Case of a Single Experiment and Finite Sample Space -- 1. Basic facts. Maximal and extremal families -- 2. Induced maximal and extremal families -- 3. Convexity, maximal and extremal families -- 4. Some examples -- II Simple Repetitive Structures of Product Type. Discrete Sample Spaces -- 0. Conditional independence -- 1. Preliminaries. Notation -- 2. Notions of sufficiency -- 3. Maximal and extremal families -- 4. Limit theorems for maximal and extremal families -- 5. The topology of $$\left( {\mathop{{\dot{U}}}\limits_{n} {{y}_{n}}} \right)UM. $$ Boltzmann laws -- 6. Integral representation of M -- 7. Construction of maximal and extremal families -- 8. On the triviality of the tail ?-algebra of a Markov chain -- 9. Examples of extremal families -- 10. Bibliographical notes -- III Repetitive Structures of Power Type. Discrete Sample Spaces -- 0. Basic facts about Abelian semigroups -- 1. Extremal families for semigroup statistics -- 2. General exponential families -- 3. The classical case.zd-valued statistics -- 4. Maximum likelihood estimation in general exponential families -- 5. Examples of general exponential families -- 6. Bibliographical notes -- IV General Repetitive Structures of Polish Spaces. Projective Statistical Fields -- 0. Probability measures on Polish spaces -- 1. Projective systems of Polish spaces and Markov kernels -- 2. Projective statistical fields -- 3. Canonical projective statistical fields on repetitive structures -- 4. Limit theorems for maximal and extremal families on repetitive structures -- 5. Poisson Models -- 6. Exponential Families -- 7. Examples from continuous time stochastic processes -- 8. Linear normal models -- 9. The Rasch model for item analysis -- 10. Bibliographical notes -- Literature.