We introduce (σ,τ)-algebras as a framework for twisted differential calculi over noncommutative, as well as commutative, algebras with motivations from the theory of σ-derivations and quantum groups. A (σ,τ)-algebra consists of an associative algebra together with a set of (σ,τ)-derivations, and corresponding notions of (σ,τ)-modules and connections are introduced. We prove that (σ,τ)-connections exist on projective modules, and introduce notions of both torsion and curvature, as well as compatibility with a hermitian form, leading to the definition of a Levi-Civita (σ,τ) connection. To illustrate the novel concepts, we consider (σ,τ) algebras and connections over matrix algebras in detail.