I am going to describe Hochschild-Witt homology, a new homology theory for associative algebras and DG algebras over a finite field that extends crystalline cohomology to the non-commutative setting. This is based on reexamination and generalization of the classical notion of Witt vectors. Using this, we introduce what we call a Hochschild-Witt complex, a gadget refining the usual Hochschild homology complex of an algebra in exactly the same way as the de Rham-Witt complex of Deligne and Illusie refines the usual de Rham complex of a smooth algebraic variety over a finite field.