Memory effects have been widely used in biology, brain science, economy, engineering and other fields. This paper will start with the mathematical definition of memory effect, explore the current models, applications and numerical simulation algorithms to describe memory effect, and analyse the difficulties of current algorithms to solve problems.
    Firstly, this paper gives the mathematical definition of memory effects by analyzing the memory effect in real life, and then analyzes and discusses the current mathematical models of memory effect: delay differential model, delay integral model, complex delay network and so on. And we discusses the relationship between different models in detail. Then, the application of memory effects in numerical calculation is discussed by using numerical algorithm to solve differential equations. Finally, we briefly discuss the current numerical algorithms for simulating memory effects and point out the difficulties they face in solving high-dimensional problems.