We study the theory of measurable projective representations for a compact quantum group \mathbb{G} , i.e., actions of \mathbb{G} on \mathcal{B}(H) for some Hilbert space H . We show that any such measurable projective representation is inner, and is hence induced by an \Omega -twisted representation for some measurable 2 -cocycle \Omega on \mathbb{G} . We show that a projective representation is continuous , i.e., restricts to an action on the compact operators \mathcal{K}(H) , if and only if the associated 2 -cocycle is regular, and that this condition is automatically satisfied if \mathbb{G} is of Kac type. This allows in particular to characterise the torsion of projective type of \widehat{\mathbb{G}} in terms of the projective representation theory of \mathbb{G} . For a given regular 2 -cocycle \Omega , we then study \Omega -twisted actions on \mathrm{C}^{*} -algebras. We define deformed crossed products with respect to \Omega , obtaining a twisted version of the Baaj–Skandalis duality and the Green–Julg isomorphism, and a quantum version of the Packer–Raeburn’s trick.