We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form

(d/dt)x(t)=A(t)x(t)+(d/dt)Q(t,x(t−g(t)))+G(t,x(t),x(t−g(t))).

In the process we use the fundamental matrix solution of

y′=A(t)y

and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation.