abstract: We extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving, to quenched random piecewise hyperbolic dynamics. For general ergodic sequences of maps in a neighbourhood of a hyperbolic map we prove a quenched large deviations principle (LDP), central limit theorem (CLT), and local central limit theorem (LCLT) ; In collaboration with D. Dragičević , G. Froyland2, C. González-Tokman.