To address various types of delays including the neutral-type arising in dynamical networks, this paper deals with coupled delay differential and continuous-time difference equations and develops stability and robustness criteria.Subsystems described by differential equations are not required to be input-to-state stable. No assumptions on network topology are made. To tackle networks in such a general formulation, this paper explicitly constructs Lyapunov-type functionals establishing stability and robustness of the overall networks. The construction requires only simple characterizations of subsystems in terms of inequalities with Lyapunov functions and instantaneous norms.