  In this paper, a new axiomatization for unbounded functional calculi is
proposed and the associated theory is elaborated comprising, among others,
uniqueness and compatibility results and extension theorems of algebraic and
topological nature. In contrast to earlier approaches, no commutativity
assumptions need to be made about the underlying algebras.
  In a second part, the abstract theory is illustrated in familiar situations
(sectorial operators, semigroup generators, normal operators). New topological
extension theorems are proved for the sectorial calculus and the
Hille--Phillips calculus. Moreover, it is shown that the Stieltjes and the
Hirsch calculus for sectorial operators are subcalculi of a (small) topological
extension of the sectorial calculus.
