  Topological invariants have proved useful for analyzing emergent function as
they characterize a property of the entire system, and are insensitive to local
details, disorder, and noise. They support edge states, which reduce the system
response to a lower dimensional space and offer a mechanism for the emergence
of global cycles within a large phase space. Topological invariants have been
heavily studied in quantum electronic systems and been observed in other
classical platforms such as mechanical lattices. However, this framework
largely describes equilibrium systems within an ordered crystalline lattice,
whereas biological systems are often strongly non-equilibrium with stochastic
components. We review recent developments in topological states in discrete
stochastic models in 1d and 2d systems, and initial progress in identifying
testable signature of topological states in molecular systems and ecology.
These models further provide simple principles for targeted dynamics in
synthetic systems or in the engineering of reconfigurable materials. Lastly, we
describe novel theoretical properties of these systems such as the necessity
for non-Hermiticity in permitting edge states, as well as new analytical tools
to reveal these properties. The emerging developments shed light on fundamental
principles for non-equilibrium systems and topological protection enabling
robust biological function.
