  Given quantities $\Delta_1,\Delta_2,\dots\geqslant 0$, a fundamental problem
in Diophantine approximation is to understand which irrational numbers $x$ have
infinitely many reduced rational approximations $a/q$ such that
$|x-a/q|<\Delta_q$. Depending on the choice of $\Delta_q$ and of $x$, this
question may be very hard. However, Duffin and Schaeffer conjectured in 1941
that if we assume a "metric" point of view, the question is governed by a
simple zero--one law: writing $\varphi$ for Euler's totient function, we either
have $\sum_{q=1}^\infty \varphi(q)\Delta_q=\infty$ and then almost all
irrational numbers (in the Lebesgue sense) are approximable, or
$\sum_{q=1}^\infty\varphi(q)\Delta_q<\infty$ and almost no irrationals are
approximable. We present the history of the Duffin--Schaeffer conjecture and
the main ideas behind the recent work of Koukoulopoulos--Maynard that settled
it.
