We start by reviewing the classical notions of Wyner's and Gács-Körner-Witsenhausen's (GKW's) common information (CI). We then segue to their extensions. For Wyner's CI, we discuss Kumar, Li and El Gamal's exact CI and show that Wyner's CI and the exact CI are special cases of a more general family of CIs known as Rényi CIs. For GKW's CI, we unveil its connections to the non-interactive correlation distillation problem, the noise-stability problem, the isoperimetric problem and hypercontractivity inequalities. We discuss several advances on these topics, including important conjectures and open problems and their partial solutions, e.g., Courtade-Kumar's conjecture, Mossel's mean-1/4 stability problem and Ordentlich-Polyanskiy-Shayevitz's conjecture. We present this “zoo” of CI measures and their connections to other problems in a unified manner. Our treatments hinge on the concept of simulation of random variables. We demonstrate that the proofs of several results in recent works (by us and other authors) can be presented in a pedagogically coherent manner. Intuitive explanations will be provided by means of illustrations and examples.