  We present and analyze the first example of a dynamical system that naturally
exhibits attracting periodic orbits that are \textit{unstable}. These unstable
attractors occur in networks of pulse-coupled oscillators where they prevail
for large networks and a wide range of parameters. They are enclosed by basins
of attraction of other attractors but are remote from their own basin volume
such that arbitrarily small noise leads to a switching among attractors.
