In the last century Geometry underwent several substantial extensions and revisions based on the fundamental revolutions that it lived through in the XIXth century.

The purpose of the lecture is to discuss several aspects of these transformations: the new concepts that emerged from these new points of view, the new perspectives that could be drawn, some bringing together the continuous and discrete viewpoints, some classical problems that could be solved, and the new interactions with other disciplines that went along.

The lecture is meant for a general mathematical audience.