We discuss non-commutative algebraic geometry in terms of differential graded (DG) categories, thought of as categories of sheaves on non-commutative spaces. We focus particularly on the Hochschild complex of a DG category with its natural S^1-action as the analogue of the de Rham complex with its de Rham differential, and discuss the Hochschild-Kostant-Rosenberg theorem which makes precise the relation between Hochschild chains and the de Rham complex. Finally, we explain how negative cylic chains/S^1-invariant chains in the Hochschild complex give rise to closed differential forms on the `moduli space of objects’ in a DG category.