  Selberg and Morris integral probability distributions are long conjectured to
be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative
Chaos measures with non-random logarithmic potentials on the unit interval and
circle, respectively. The construction and properties of these distributions
are reviewed from three perspectives: analytic based on several representations
of the Mellin transform, asymptotic based on low intermittency expansions, and
probabilistic based on the theory of Barnes beta probability distributions. In
particular, positive and negative integer moments, infinite factorizations and
involution invariance of the Mellin transform, analytic and probabilistic
proofs of infinite divisibility of the logarithm, factorizations into products
of Barnes beta distributions, and Stieltjes moment problems of these
distributions are presented in detail. Applications are given in the form of
conjectured mod-Gaussian limit theorems, laws of derivative martingales,
distribution of extrema of $1/f$ noises, and calculations of inverse
participation ratios in the Fyodorov-Bouchaud model.
