  We introduce the concept of pseudotwistor (with particular cases called
twistor and braided twistor) for an algebra $(A, \mu, u)$ in a monoidal
category, as a morphism $T:A\otimes A\to A\otimes A$ satisfying a list of
axioms ensuring that $(A, \mu \circ T, u)$ is also an algebra in the category.
This concept provides a unifying framework for various deformed (or twisted)
algebras from the literature, such as twisted tensor products of algebras,
twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors
appear also in other topics from the literature, e.g. Durdevich's braided
quantum groups and ribbon algebras. We also focus on the effect of twistors on
the universal first order differential calculus, as well as on lifting twistors
to braided twistors on the algebra of universal differential forms.
