In this talk, we discuss various aspects of twice epi-differentiablity of extended-real-valued functions and demonstrate how it can be leveraged to study the local convergence analysis of the augmented Lagrangian method for composite optimization problems. We begin with presenting the history of twice epi-differentiablity and proceed with its evolution in the last three decades. In particular, we demonstrate that this property often holds for various classes of functions, commonly seen in applications to optimization problems. If time allows, we will discuss further applications of this concept in stability properties of generalized equations and show that it can be used to achieve a simple characterization of continuous differentiability of the proximal mapping for a large class of functions, called C^2-decomposable.