Geometry of Banach Spaces Duality Mappings and Nonlinear Problems I. Subdifferentiability and Duality Mappings § 1. Generalities on convex functions § 2. The subdifferential and the conjugate of a convex function § 3. Smooth Banach spaces § 4. Duality mappings on Banach spaces § 5. Positive duality mappings Exercises Bibliographical comments II Characterizations of Some Classes of Banach Spaces by Duality Mappings § 1. Strictly convex Banach spaces § 2. Uniformly convex Banach spaces § 3. Duality mappings in reflexive Banach spaces § 4. Duality mappings in LP spaces § 5. Duality mappings in Banach spaces with the property (h) and ()1 Exercises Bibliographical comments III Renorming of Banach Spaces § 1. Classical renorming results § 2. Lindenstrauss’ and Trojanski’s Theorems Exercises Bibliographical comments IV On the Topological Degree in Finite and Infinite Dimensions § 1. Brouwer’s degree § 2. Browder Petryshyn’s degree for A proper mappings § 3. P compact mappings Exercises Bibliographical comments V Nonlinear Monotone Mappings § 1. Demicontinuity and hemicontinuity for monotone operators § 2. Monotone and maximal monotone mappings § 3. The role of the duality mapping in surjectivity and maximality problems § 4. Again on subdifferentials of convex functions Exercises Bibliographical comments VI Accretive Mappings and Semigroups of Nonlinear Contractions § 1. General properties of maximal accretive mappings § 2. Semigroups of nonlinear contractions in uniformly convex Banach spaces § 3. The exponential formula of Crandall Liggett § 4. The abstract Cauchy problem for accretive mappings § 5. Semigroups of nonlinear contractions in Hilbert spaces § 6. The inhomogeneous case Exercises Bibliographical comments References.