  The Bernoulli sieve is a version of the classical balls-in-boxes occupancy
scheme, in which random frequencies of infinitely many boxes are produced by a
multiplicative random walk, also known as the residual allocation model or
stick-breaking. We give an overview of the limit theorems concerning the number
of boxes occupied by some balls out of the first $n$ balls thrown, and present
some new results concerning the number of empty boxes within the occupancy
range.
