  We establish a comprehensive local wellposedness theory for the quasilinear
Maxwell system with interfaces in the space of piecewise $H^m$-functions for $m
\geq 3$. The system is equipped with instantaneous and piecewise regular
material laws and perfectly conducting interfaces and boundaries. We also
provide a blow-up criterion in the Lipschitz norm and prove the continuous
dependence on the data. The proof relies on precise a priori estimates and the
regularity theory for the corresponding linear problem also shown here.
