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A k-differential is a section of the k-th power of the cotangent bundle on a Riemann surface. The space of k-differentials is stratified by prescribing the number and multiplicities of their zeros and poles. The cases of abelian and quadratic differentials, corresponding to k=1 and k=2 respectively, exhibit fascinating geometry that arises from the associated flat structures, which provide us a good understanding of the dimension, local coordinates, connected components, and compactification for the strata. In this talk I will report some recent work in progress that studies these aspects of the strata of k-differentials for general k. It is part of a joint project with Bainbridge, Gendron, Grushevsky, and Moeller.