Periodic motions in DDE (Differential-Delay Equations) are typically created in Hopf bifurcations. In this talk we examine this process from several points of view. We begin with an application to gene copying in which the delay is due to an observed time lag in the transcription process. Then we use Lindstedt's perturbation method to derive the DDE Hopf Bifurcation Formula, which determines the stability of the periodic motion. Next we use Center Manifold Analysis to reduce the DDE from an an infinite dimensional evolution equation on a function space to a two dimensional ODE (Ordinary Differential Equation) on the center manifold, the latter being a surface tangent to the eigenspace associated with the Hopf bifurcation. We close with an example based on two coupled delay limit cycle oscillators.