The pentagram map is a beautiful discrete, completely integrable system with many relations to other mathematical domains. It was originally defined by R.Schwartz in 1992 as a map on plane convex polygons, where a new polygon is spanned by the “shortest” diagonals of the initial one. We describe various extensions and the geometry of this map in higher dimensions, and in particular the recently found long-diagonal maps encompassing known integrable cases. We also describe the corresponding continuous limits of such maps, which happen to coincide with equations of the KdV hierarchy, generalizing the Boussinesq equation in 2D. This is a joint work with Fedor Soloviev and Anton Izosimov.