Emil Wiedemann, Hausdorff Center for Mathematics, Bonn, Germany Joint work with G. Crippa, N. Gusev, and S. Spirito In a seminal work from 1989, DiPerna and Lions defined the notion of renormalized solution for linear continuity equations. They showed that there exists, for every initial data, a unique renormalized solution. On the other hand, if the regularity of the transporting vector field is sufficiently low, examples are known of non-renormalized distributional solutions. I will present recent results that allow to construct non-renormalized solutions to continuity equations with an essentially arbitrary renormalization defect, using a convex integration-type argument.