  We introduce and study the definition, main properties and applications of
iterated twisted tensor products of algebras, motivated by the problem of
defining a suitable representative for the product of spaces in noncommutative
geometry. We find conditions for constructing an iterated product of three
factors, and prove that they are enough for building an iterated product of any
number of factors. As an example of the geometrical aspects of our
construction, we show how to construct differential forms and involutions on
iterated products starting from the corresponding structures on the factors,
and give some examples of algebras that can be described within our theory. We
prove a certain result (called ``invariance under twisting'') for a twisted
tensor product of two algebras, stating that the twisted tensor product does
not change when we apply certain kind of deformation. Under certain conditions,
this invariance can be iterated, containing as particular cases a number of
independent and previously unrelated results from Hopf algebra theory.
