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We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on ${ℝ}^d$, equipped with power weights $w\left(x\right)={\left|x\right|}^{\gamma }$, γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.