The works collected in this volume, while addressing quite different goals, are focused on the same type of mathematical object: cocycles over partially hyperbolic diffeomorphisms. We begin with a preliminary overview giving background on the history and applications of the study of dynamical cocycles and partially hyperbolic theory and elucidating the connections between the two main articles. The first one investigates effective conditions which ensure that the Lyapunov spectrum of a (possibly non-linear) cocycle over a partially hyperbolic dynamical system is nontrivial. In the second one, the classical Livšic theory of the cohomological equation for Anosov diffeomorphisms is extended to accessible partially hyperbolic diffeomorphisms.