Abstract: While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace--Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes.