Systems of partial differential equations (PDE) and ordinary differential equations (ODE) are studied from the point of view of Dynamical systems methods. Many problems from biology, chemistry, mechanics, control, information transmission, economics and other fields are changing in time and so they can be described by different types of Dynamical systems. It is well understood that taking into account delay effects (memory) makes the mathematical models more realistic. We are interested in developing approaches for study of local and long-time asymptotic behavior of different types of solutions to delay PDEs and ODEs. An important type of delay is the state-dependent one. This type of delay seems to be the most natural from the point of view of applications (it also includes the case of constant delay) and simultaneously the most difficult from mathematical point of view. The main goal of the project is therefore to study basic properties of such systems, including a new type of dynamical state-dependent delay and coupled differential-algebraic systems of delay equations.