Is it possible to distinguish algebraic from transcendental real numbers by
considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel
suggested that the answer is no and that for any real irrational algebraic
number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should
satisfy some of the laws that are shared by almost all numbers. There is no
explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer,
$a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for
which one can claim that the digit $a$ occurs infinitely often in the $g$-ary
expansion of $x$. However, some progress has been made recently, thanks mainly
to clever use of Schmidt's subspace theorem. We review some of these results.