Delay differential equations arise in many applications, and in the case of constant delays solutions give rise to semi-flows on function spaces. A fairly mature theory of such problems as infinite dimensional dynamical systems has now been developed. However, models in physical and biological applications are increasingly encompassing features which do not fit this theory, often having non-constant and state-dependent delays. Mixed type differential equations with both advanced and retarded arguments also arise, for example as the defining equations for travelling waves in nonlinear lattices. Volterra functional (integral and integro-differential) equations with variable and state-dependent delays, including equations of “integral-algebraic” type, are also used with increasing frequency. The theory of such problems is still not complete, though significant progress has been made in recent years. A large gap also exists in the numerical analysis and computational solution of such functional equations.