  This paper generalizes Wyner's definition of common information of a pair of
random variables to that of $N$ random variables. We prove coding theorems that
show the same operational meanings for the common information of two random
variables generalize to that of $N$ random variables. As a byproduct of our
proof, we show that the Gray-Wyner source coding network can be generalized to
$N$ source squences with $N$ decoders. We also establish a monotone property of
Wyner's common information which is in contrast to other notions of the common
information, specifically Shannon's mutual information and G\'{a}cs and
K\"{o}rner's common randomness. Examples about the computation of Wyner's
common information of $N$ random variables are also given.
