The talk begins with examples of differential equations with time lags, which are state-dependent or time dependent. These range from prototypic models for automatic soft landing to models for populations of dividing cells. A toy example shows that equations with variable time lags require a framework which is different from the theory of functional differential equations developed in the last century. For autonomous equations with bounded delay we provide basic results on well-posedness of initial value problems, which among others resolve the former problem of linearization for equations with state-dependent delay, and discuss results and open problems for more general equations: Algebraic-delay systems, neutral equations, cases of unbounded delay, and nonautonomous equations. We report about case studies and present a most recent result which shows what state-dependent delay can do if introduced in the simplest linear delay differential equation.