This chapter introduces functional differential equations (FDEs), which are categorized into delay differential equations (DDEs), neutral FDEs (NFDEs), and mixed FDEs (MFDEs). The classification depends on how the current change rate of the system state depends on its history or future expectations. DDEs with state-dependent delays are also discussed. The chapter then delves into infinite dynamical systems generated by time lags, using Newtonian mechanics as an example. In Newtonian mechanics, the system's state changes over time according to an ordinary differential equation (ODE). However, in many applications, the change rate depends not only on the current state but also on the system's history, leading to DDEs of the form: \[ \dot{x}(t) = f(x(t), x(t - \tau)), \] where \( x(t) \) is the system's state at time \( t \), \( f \) is a given mapping, and \( \tau \) is a constant time lag. This equation naturally arises in population dynamics, such as the maturation of a single-species structured population. To solve such equations, the initial history of the system must be prescribed on the interval \([- \tau, 0]\). The initial value problem: \[ \dot{x}(t) = f(x(t), \varphi(t - \tau)), \quad t \in [0, \tau], \, x(0) = \varphi(0), \] is solvable if \( f \) is continuous and locally Lipschitz with respect to the first state variable \( x \). This process can be repeated to find solutions over longer intervals, leading to a semiflow on the Banach space \( C_{n, \tau} \) of continuous mappings from \([- \tau, 0]\) to \( \mathbb{R}^n \). The study of asymptotic behaviors of solutions to DDEs is challenging due to the infinite-dimensionality of the phase space and the generated semiflow.This chapter introduces functional differential equations (FDEs), which are categorized into delay differential equations (DDEs), neutral FDEs (NFDEs), and mixed FDEs (MFDEs). The classification depends on how the current change rate of the system state depends on its history or future expectations. DDEs with state-dependent delays are also discussed. The chapter then delves into infinite dynamical systems generated by time lags, using Newtonian mechanics as an example. In Newtonian mechanics, the system's state changes over time according to an ordinary differential equation (ODE). However, in many applications, the change rate depends not only on the current state but also on the system's history, leading to DDEs of the form: \[ \dot{x}(t) = f(x(t), x(t - \tau)), \] where \( x(t) \) is the system's state at time \( t \), \( f \) is a given mapping, and \( \tau \) is a constant time lag. This equation naturally arises in population dynamics, such as the maturation of a single-species structured population. To solve such equations, the initial history of the system must be prescribed on the interval \([- \tau, 0]\). The initial value problem: \[ \dot{x}(t) = f(x(t), \varphi(t - \tau)), \quad t \in [0, \tau], \, x(0) = \varphi(0), \] is solvable if \( f \) is continuous and locally Lipschitz with respect to the first state variable \( x \). This process can be repeated to find solutions over longer intervals, leading to a semiflow on the Banach space \( C_{n, \tau} \) of continuous mappings from \([- \tau, 0]\) to \( \mathbb{R}^n \). The study of asymptotic behaviors of solutions to DDEs is challenging due to the infinite-dimensionality of the phase space and the generated semiflow.