Delay differential equations are wily beasts. Though they share much of their DNA with ordinary differential equations, their dependence on past states introduces infinite-dimensional subtleties that complicate their analysis. In this thesis, we explore some of the many challenges that arise in studying nonlinear oscillations in delay differential equations, along with techniques we have found useful for taming them. We begin with an overview of the foundational theory, followed by thoroughly worked examples from the linear case. We then turn to nonlinear oscillations, focusing on the deceptively simple Delayed Duffing Equation. Several approaches to this equation are presented, including the Fourier-based Method of Harmonic Balance. Next, we introduce a new technique for connecting solutions of delay equations to their undelayed counterparts, which we call the Method of Characteristic Oscillations. Finally, we take on the formidable problem of chaos in delay systems. Our hope is that by the end of this thesis, the reader will feel well equipped to venture into the wild and fascinating world of delay equations.