A large number of complex systems, naturally emerging in various domains, are well described by directed networks, resulting in numerous interesting features absent from their undirected counterparts. Among these properties is a strong non-normality, inherited by a strong asymmetry that characterizes such systems and guides their underlying hierarchy. Dynamical processes evolving on non-normal networks exhibit peculiar behavior: initial small disturbances of stable systems may undergo a transient growth phase in the linear regime that can eventually transform to a permanent instability in the nonlinear one. In this talk, we present recent findings related to emergent structural and dynamical properties of non-normal networks. In particular, we will first show using information-theoretic tools that a ubiquitous feature is observed amongst such systems as the level of non-normality increases. When the non-normality reaches a given threshold, highly directed substructures aiming towards terminal (sink or source) nodes, denoted here as leaders, spontaneously emerge. Such hierarchical features trigger many collective behaviors, where in particular, we will focus on the synchronization phenomenon. We demonstrate that contrary to the common belief that strong directedness ensures the robustness of synchronization, due to a transient growth induced by the structure’s non-normality, the system might lose synchronization.