Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundamental properties and isolate some new ones, including a characterisation of the limit superior and inferior of sequences as limits along tensor products, and an application to the Banach asymptotic density. We then prove a general result on the combinatorial structure of sets belonging to tensor products and, as a result, we obtain several characterisations of the additive properties of sets of natural numbers. Finally, we show that tensor products can be described as idempotents of an appropriate semigroup.