Limit theorems are at the core of probability theory as they offer key information on how sums of random variables and stochastic processes behave. From the classical Law of Large Numbers (LLN) and Central Limit Theorem (CLT) more recent generalizations with heavy tails, random environments or dependent structures among test variables, duality between theory development and practical application is a characteristic feature of the progress in limit theorems.

The chapter discusses various important new developments in limit theorems, including generalizations of classical results, relationship between these and stable laws and heavy-tails phenomena, functional limit theorems and applications to a number of areas, such as finance, network traffic analysis or statistical physics up to machine learning and stochastic modelling of complex systems. We explore into recent advances nonclassical limits, refined convergence modes and large deviation principles, high dimensional probability which demonstrates modern probability combines theory with applications. The chapter ends with a presentation of possible future work and a selection of references for further study.