O for orientation: which geometry is more convenient?

Geometry is an ancient discipline dealing with mathematical shapes and patterns, which the ancient Greeks, especially Plato, called “ideas”. In the nineteenth century, when idealism flourished again, geometry went through a phase of re-foundation that made it a modern system of thought, fundamental to cultural and technological development. Through the drafting of programmes and manifestos, typical of that century, it came out of the narrow Euclidean environment and offered mathematicians greater prospects and different choices to interpret the space around us (or in which we address our problems). In this paper, I present some contemporary geometric research programmes, in mathematics as well as in other disciplines, their state of advancement and their applications.