A k-differential on a Riemann surface is a section of the k-th power of the canonical line bundle. Loci of k-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of k-differentials. In this paper we give a complete description for the compactification of the strata of k-differentials in terms of pointed stable k-differentials, for all k. The upshot is a global k-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of k-differentials regarding their deformations, residues, and flat geometric structure.