Understanding the relationship between the structure of sparse random networks and their spectra is one of the central and long-standing focus of network theory. Spectral properties of networks determine the performance of algorithms, phase transitions, and the linear stability of complex biological systems, such as neural networks and ecosystems. In this talk, I will present simple analytic results for the leading eigenvalue of directed random networks and for the statistics of the corresponding leading eigenvector. Specifically, the leading eigenvalue undergoes a gap-gapless transition as a function of the network parameters, while the leading eigenvector may become localized. These results provide a clear understanding of the impact of the network structure on the leading eigenpair of directed networks, which is relevant for characterizing the phase diagram of nonlinear complex systems with sparse interactions. Finally, I will show that even in highly connected networks, moderate degree fluctuations can lead to a breakdown of Gaussian random-matrix universality.