  A central problem of geometry is the tiling of space with simple structures.
The classical solutions, such as triangles, squares, and hexagons in the plane
and cubes and other polyhedra in three-dimensional space are built with sharp
corners and flat faces. However, many tilings in Nature are characterized by
shapes with curved edges, non-flat faces, and few, if any, sharp corners. An
important question is then to relate prototypical sharp tilings to softer
natural shapes. Here, we solve this problem by introducing a new class of
shapes, the \textit{soft cells}, minimizing the number of sharp corners and
filling space as \emph{soft tilings}. We prove that an infinite class of
polyhedral tilings can be smoothly deformed into soft tilings and we construct
the soft versions of all Dirichlet-Voronoi cells associated with point lattices
in two and three dimensions. Remarkably, these ideal soft shapes, born out of
geometry, are found abundantly in nature, from cells to shells.
