  There is enormous interest -- both mathematically and in diverse applications
-- in understanding the dynamics of coupled oscillator networks. The real-world
motivation of such networks arises from studies of the brain, the heart,
ecology, and more. It is common to describe the rich emergent behavior in these
systems in terms of complex patterns of network activity that reflect both the
connectivity and the nonlinear dynamics of the network components. Such
behavior is often organized around phase-locked periodic states and their
instabilities. However, the explicit calculation of periodic orbits in
nonlinear systems (even in low dimensions) is notoriously hard, so
network-level insights often require the numerical construction of some
underlying periodic component. In this paper, we review powerful techniques for
studying coupled oscillator networks. We discuss phase reductions,
phase-amplitude reductions, and the master stability function for smooth
dynamical systems. We then focus in particular on the augmentation of these
methods to analyze piecewise-linear systems, for which one can readily
construct periodic orbits. This yields useful insights into network behavior,
but the cost is that one needs to study nonsmooth dynamical systems. The study
of nonsmooth systems is well-developed when focusing on the interacting units
(i.e., at the node level) of a system, and we give a detailed presentation of
how to use \textit{saltation operators}, which can treat the propagation of
perturbations through switching manifolds, to understand dynamics and
bifurcations at the network level. We illustrate this merger of tools and
techniques from network science and nonsmooth dynamical systems with
applications to neural systems, cardiac systems, networks of electro-mechanical
oscillators, and cooperation in cattle herds.
