This paper is the study of nontrivial periodic solutions of parametrized families of differential-delay equations of the form \[ (1)\quad \dot y(t)=h_{\lambda}(y(t-N_ 1),y(t-N_ 2),...,y(t-N_ k)), \] where \(h_{\lambda}: R^ k\to R\) is a continuous map and \(N_ 1,N_ 2,...,N_ k\) are nonnegative integers. The general approach of the author in this paper is as follows: If one assumes that y(t) is a periodic solution of equation (1) of integral period p and one defines \(u_ j(t)=y(t-j+1)\) for \(1\leq j\leq p\), one discovers that \(u(t)=(u_ 1(t),...,u_ p(t))\) satisfies a ”cyclic system of ordinary differential equations”, say (2) \(\dot u(t)=g(u(t)).\) The results deal with proving that equation (1) has no periodic solutions of period p basically by proving that every solution of equation (2) satisfies \(\lim_{t\to \infty}| u(t)| =0\) or \(\lim_{t\to \infty}| u(t)| =\infty,\) and the key step in proving the latter statement is the construction of appropriate Lyapunov functions for equation (2).