  There have been several recent articles studying homology of various types of
random simplicial complexes. Several theorems have concerned thresholds for
vanishing of homology, and in some cases expectations of the Betti numbers.
However little seems known so far about limiting distributions of random Betti
numbers.
  In this article we establish Poisson and normal approximation theorems for
Betti numbers of different kinds of random simplicial complex:
Erd\H{o}s-R\'enyi random clique complexes, random Vietoris-Rips complexes, and
random \v{C}ech complexes. These results may be of practical interest in
topological data analysis.
