Let $\beta>1$. A representation of a number $x\in[0,1]$ in the

form \[ x = \sum_{r=0}^\infty \frac{d_r}{\beta^{r+1}},\] where the digits
$d_r$ are non-negative integers less than $\beta$, is called a {\em
$\beta$-expansion} of $x$. If $\beta$ is not an integer, then almost every
$x$ has uncountably many distinct $\beta$-expansions, but there is a
preferred choice, the {\em greedy} $\beta$-expansion (first studied by
R\'enyi in 1957) for which the digit sequence is lexicographically largest.

The talk will consist of a general survey of the properties of
$\beta$-expansions, followed by a description of some recent work with
Boyland and de Carvalho in which we describe, for each $\beta$, the set of
possible digit frequencies in greedy $\beta$-expansions of numbers
$x\in[0,1]$.