Suppose J=[:,
) for some : # R or J=R and let X be a Banach space. We study asymptotic behavior of solutions on J of neutral system of equations with values in X. This reduces to questions concerning the behavior of solutions of convolution equations (*) H V 0=b, where H=(H j, k ) is an r_r matrix, H j, k # D$ L 1 , b=(b j ) and b j # D$(R, X ), for 1 j, k r. We prove that if 0 is a bounded uniformly continuous solution of (*) with b from some translation invariant suitably closed class A, then 0 belongs to A, provided, for example, that det H has countably many zeros on R and c 0 / 3 X. In particular, if b is (asymptotically) almost periodic, almost automorphic or recurrent, 0 is too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equation 0(t+1)&0(t)=b(t) with J=X=R and b Stepanoff almost periodic.