Hilbert geometry, defined on any convex body in a real affine space, is a rich source of examples of metric spaces and has had numerous applications since its description by Hilbert in 1895. The members of this consortium are contributing to various generalizations of this concept and its applications to different contexts, of affine spaces over other field than the real numbers. The objective of this project is threefold: - to develop a unified approach to these generalizations: unified definitions, common generalization of Benzecri's results and of notions of volumes; - to explore the interplay between the different contexts, through numerous examples; - to obtain meaningful applications of Hilbert geometry in each specific case. Applications include projects around: - the study of the metrics of minimal entropy for symmetric spaces; - degeneracy of convex projective structures on surfaces; - around the frontier of the set of Anosov representations in complex hyperbolic geometry; - new linear programming algorithms, with Smale's 9th problem in mind.