Abstract. Real-world networks in technology, engineering and
biology often exhibit dynamics that cannot be adequately reproduced
using network models given by smooth dynamical systems
and a fixed network topology. Asynchronous networks give a theoretical
and conceptual framework for the study of network dynamics
where nodes can evolve independently of one another, be
constrained, stop, and later restart, and where the interaction between
different components of the network may depend on time,
state, and stochastic effects. This framework is sufficiently general
to encompass a wide range of applications ranging from engineering
to neuroscience. Typically, dynamics is piecewise smooth and
there are relationships with Filippov systems. In the first part of
the paper, we give examples of asynchronous networks, and describe
the basic formalism and structure. In the second part, we
make the notion of a functional asynchronous network rigorous,
discuss the phenomenon of dynamical locks, and present a foundational
result on the spatiotemporal factorization of the dynamics
for a large class of functional asynchronous networks.