  We uncover a topological classification applicable to open fermionic systems
governed by a general class of Lindblad master equations. These `quadratic
Lindbladians' can be captured by a non-Hermitian single-particle matrix which
describes internal dynamics as well as system-environment coupling. We show
that this matrix must belong to one of ten non-Hermitian Bernard-LeClair
symmetry classes which reduce to the Altland-Zirnbauer classes in the closed
limit. The Lindblad spectrum admits a topological classification, which we show
results in gapless edge excitations with finite lifetimes. Unlike previous
studies of purely Hamiltonian or purely dissipative evolution, these
topological edge modes are unconnected to the form of the steady state. We
provide one-dimensional examples where the addition of dissipators can either
preserve or destroy the closed classification of a model, highlighting the
sensitivity of topological properties to details of the system-environment
coupling.
