Delay differential equations (DDEs) are a type of differential equation that involve delays in the derivative of the unknown function. In DDEs, the rate of change of a function at a given time depends not only on its value at that time, but also on its values at previous times. This time delay can arise from various factors, such as time for information to propagate through a system or finite speeds of various processes. DDEs are commonly used in modeling various real-world phenomena in biology, chemistry, physics, and engineering, where time delays play a crucial role in the dynamics of the system. The study of DDEs involves analyzing the stability and behavior of solutions to these equations, as well as developing numerical methods for their solution. Overall, delay differential equations provide a powerful tool for understanding systems with time delays and capturing the complex dynamics that emerge from these delays.