The author studies the existence of bounded solutions of the equation \[ (1)\quad u'(t)=Au(t)+\int^{\infty}_{0}dB(\tau)u(t-\tau)+f(t),\quad t\in {\mathbb{R}}, \] in a Banach space X. Here A is a closed linear operator with dense domain D(A) and \(B\in BV({\mathbb{R}}_+,B(D(A),X))\). The main problem is which spaces of bounded continuous functions are admissible for (1), i.e., such that for f in this space there exists a unique solution that depends continuously on f. A number of characterizations of admissible spaces are given, mainly in terms of the spectrum of f and of the Laplace (or Fourier) transform, of d/dt-A-dB. Almost periodic and asymptotically almost periodic functions are given special consideration. There is some trade-off between restrictions of f and assumptions on A, B, and the space X. These results are applied to the study of the resolvent associated with the equation \(u'(t)=Au(t)+\int^{t}_{0}dB(\tau)u(t-\tau)+g(t),\) \(t\in {\mathbb{R}}_+\). A number of applications involving partial differential operators are given.