A delay equation is a rule for extending a function of time towards the future, on the basis of the known past. Renewal Equations prescribe the current value, while Delay Differential Equations prescribe the derivative of the current value. With a delay equation one can associate a dynamical system by translation along the extended function.

I will illustrate by way of examples how such equations arise in the description of the dynamics of structured populations and sketch the available theory, while noting the need for numerical bifurcation tools.

The lecture is based on joint work with Mats Gyllenberg, Hans Metz and many others.