We investigate the asymptotic properties of the inhomogeneous
nonautonomous evolution equation $(d/dt)u(t)= Au(t)+ B(t)u(t)+ f(t),t\in\mathbb{R}$ , where $(A,D(A))$ is a Hille-Yosida operator on a Banach space $X,B(t),t\in\mathbb{R}$ , is a family of operators in $\mathcal{L}(\overline{D(A)},X)$ satisfying certain boundedness and measurability conditions and $f\in L_{\mathrm{loc}^1(\mathbb{R},X)$ . The solutions of the corresponding homogeneous equations are represented by an evolution family $(U_B(t,s))_{t\geq s}$ . For various function spaces $\mathcal{F}$ we show conditions on $(U_B(t,s))_{t\geq s}$ and $f$ which ensure the existence of a unique solution contained in $\mathcal{F}$ . In particular, if $(U_B(t,s))_{t\geq s}$ is $p$ -periodic there exists a unique bounded solution $u$ subject to certain spectral assumptions on on $U_B(p,0),f$ and $u$ . We apply the results to nonautonomous semilinear retarded differential equations. For certain $p$ -periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of $(U_B(t,s))_{t\geq s}$ .