  Common information (CI) is ubiquitous in information theory and related areas
such as theoretical computer science and discrete probability. However, because
there are multiple notions of CI, a unified understanding of the deep
interconnections between them is lacking. This monograph seeks to fill this gap
by leveraging a small set of mathematical techniques that are applicable across
seemingly disparate problems.
  In Part I, we review the operational tasks and properties associated with
Wyner's and G\'acs-K\"orner-Witsenhausen's (GKW's) CI. In PartII, we discuss
extensions of the former from the perspective of distributed source simulation.
This includes the R\'enyi CI which forms a bridge between Wyner's CI and the
exact CI. Via a surprising equivalence between the R\'enyi CI of order~$\infty$
and the exact CI, we demonstrate the existence of a joint source in which the
exact CI strictly exceeds Wyner's CI. Other closely related topics discussed in
Part II include the channel synthesis problem and the connection of Wyner's and
exact CI to the nonnegative rank of matrices.
  In Part III, we examine GKW's CI with a more refined lens via the noise
stability or NICD problem in which we quantify the agreement probability of
extracted bits from a bivariate source. We then extend this to the $k$-user
NICD and $q$-stability problems, and discuss various conjectures in information
theory and discrete probability, such as the Courtade-Kumar, Li-M\'edard and
Mossell-O'Donnell conjectures. Finally, we consider hypercontractivity and
Brascamp-Lieb inequalities, which further generalize noise stability via
replacing the Boolean functions therein by nonnnegative functions. The key
ideas behind the proofs in Part III can be presented in a pedagogically
coherent manner and unified via information-theoretic and Fourier-analytic
methods.
