Using the evolution semigroup approach, the authors give a characterization of exponential stability for the nonautonomous partial functional-differential equation \[ \dot{u}(t) = A(t) u(t) + L(t) u_t,\quad t \geq s, \qquad u(t) = \varphi(t-s),\quad s-r \leq t \leq s, \] in terms of a generalized characteristic equation which is formulated on adequate function spaces. Here, \(A\) generates a \(C_0\)-semigroup on a Banach space \(X\). Further, \(r \geq 0\), \(\varphi \in E := C([-r,0],X)\), \(L \in {\mathcal L}(E,X)\), and \(u_t(\xi) := u(t+\xi)\) for \(\xi \in [-r,0]\), \(t \geq 0\), and \(u : [-r,\infty) \rightarrow X\). The characteristic equation is a spectral condition extending known characteristic equations for the autonomous or periodic case. The authors also deduce robustness results.