In this lecture, we introduce a family of delay-differential equations with many applications in various fields such as economics, population dynamics and physiological systems. We present some new results on the global dynamics of this equation, focusing on a particular but representative example.  Bifurcation diagrams using relevant model parameters show some interesting features, such as stability switches and extinction windows due to sudden collapses. For the general case, we also state and outline the proof of a sharp delay-independent global stability result, showing that it works for our main examples.  The interplay between the continuous dynamical system generated by the delay equation and an associated  one-dimensional discrete dynamical system plays an essential role in our approach.