We will consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. In particular, we show how many biological oscillators --- both "pulse-coupled" and "phase-coupled" --- fall into this framework. We also examine a non-standard spectral problem that generically appears in these problems.