Probabilistic formulation of the problem -- The RG-transformation for the hierarchical model -- The existence of a non-trivial fixed point -- The flow around the fixed point -- Discussion of the critical indices -- Global properties of the flow -- Discussion of the thermodyamic limit -- Perturbation theory -- Explicit perturbation calculation for the Eigenvalue ?2 -- Linear problems and the existence of ?? -- Differentiability of ø? -- The normal form of the flow -- Crossover. Part 2 -- Crossover, Part II. A) Single phase region -- Crossover Part II. B) Two phase region -- Miscellaneous short proofs.