Tensors often have too many entries to be stored explicitly so it is essential to compress them into data sparse formats. I will identify three methodologies that can be used to explain when a tensor is compressible. Each methodology leads to bounds on the compressibility of certain tensors, partially explaining the abundance of low-rank tensors in applied mathematics. In particular, I will focus on tensors with a so-called displacement structure, showing that solutions to Poisson equations on tensor-product geometries are highly compressible. As the rank bounds are constructive, I will develop an optimal-complexity spectrally-accurate 3D Poisson solver with O(n(\log n)^2(log 1/epsilon)^2) complexity for a smooth righthand side, where nxnxn is the tensor discretization of the solution.