We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any C*-algebraic bundle which is second countable and which satisfies a certain regularity condition (automatically satisfied if the unit fibre algebra is stable), there is a continuous twisted partial action of the base group on the unit fibre algebra, whose associated semidirect product bundle is isomorphic to the given one.

1991 Mathematics Subject Classification: 46L05, 46L40, 46L55.