  We analyse a collection of empirical networks in a wide spectrum of
disciplines and show that strong non-normality is ubiquitous in network
science. Dynamical processes evolving on non-normal networks exhibit a peculiar
behaviour, as initial small disturbances may undergo a transient phase and be
strongly amplified in linearly stable systems. Additionally, eigenvalues may
become extremely sensible to noise, and have a diminished physical meaning. We
identify structural properties of networks that are associated to non-normality
and propose simple models to generate networks with a tuneable level of
non-normality. We also show the potential use of a variety of metrics capturing
different aspects of non-normality, and propose their potential use in the
context of the stability of complex ecosystems.
