The author is investigating the existence of periodic solutions to the integrodifferential equation of Volterra type \[ (1)\quad x'(t)=h(t,x(t))+\int^{t}_{-\infty}q(t,s,x(s))ds, \] under the basic assumptions that \(h: R\times R^ n\to R^ n\), and \(q: R\times R\times R^ n\to R^ n\) are both continuous, h is periodic in t with period T, and \(q(t+T,s+T,x)=q(t,s,x)\) in the whole domain of definitions of q. Regarding (1) as an equation with infinite delay, the space of initial functions is chosen to be the space of continuous and bounded functions on \(R_-\), with values in \(R^ n\). Using fixed point results, and considering the systems on compact intervals \(y'(t)=h(t,y(t))+\int^{t}_{t-kT}q(t,s,y(s))ds\), \(k=1,2,...\), existence of periodic solutions is obtained under further assumptions. Uniform boundedness of solutions (not necessarily periodic!) is also obtained. Especially nonlinear cases are then discussed.