A twisted tensor product of two algebras is a non-commutative generalization of their usual tensor product, and a deformation of an algebra is a redefinition of its multiplicative law. In many interesting cases, the same algebra can be obtained via both a twisted tensor product and a deformation. In this talk we will present the canonical tool for understanding deformations, Hochschild cohomology, as well as techniques that give explicit formulas to compute it for twisted tensor products. This will include an unpretentious introduction to all objects of interest, a bountiful of examples, and the unexpected generality of our results.