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Abstract We define a local homomorphism$$(Q,k)\to (R,\ell )$$to be Koszul if its derived fiber$$R\otimes ^{\mathsf {L}}_Q k$$is formal, and if$$\operatorname {Tor}^{Q}(R,k)$$is Koszul in the classical sense. This recovers the classical definition whenQis a field, and more generally includes all flat deformations of Koszul algebras. The non-flat case is significantly more interesting, and there is no need for examples to be quadratic: all complete intersection and all Golod quotients are Koszul homomorphisms. We show that the class of Koszul homomorphisms enjoys excellent homological properties, and we give many more examples, especially various monomial and Gorenstein examples. We then study Koszul homomorphisms from the perspective of$$\mathrm {A}_{\infty }$$-structures on resolutions. We use this machinery to construct universal free resolutions ofR-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of anR-moduleMis often minimal and can be described by a finite amount of data wheneverMandRhave finite projective dimension overQ. Our construction simultaneously recovers the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring, and produces analogous resolutions for various other classes of local rings.