Abstract: The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plücker coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polytopes and hyperplane arrangements, and the positive Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field theory N=4 super Yang-Mills, and calculating this volume involves decomposing (or tiling) the amplituhedron into smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.