  A fundamental problem in contemporary string/M theory is to count the number
of inequivalent vacua satisfying constraints in a string theory model. This
article contains the first rigorous results on the number and distribution of
supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau
3-fold $X$ with flux. In particular, complete proofs of the counting formulas
in Ashok-Douglas and Denef-Douglas are given, together with van der Corput
style remainder estimates. We also give evidence that the number of vacua
satisfying the tadpole constraint in regions of bounded curvature in moduli
space is of exponential growth in $b_3(X)$.
