  We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$
linear forms in $n$ variables. That is, we establish a criterion to determine
whether, for a given rate of approximation, almost all or almost no $n$-by-$m$
systems of linear forms are approximable at that rate using integer vectors
satisfying a natural coprimality condition. When $m=n=1$, this is the classical
1941 Duffin--Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos
and Maynard. Pollington and Vaughan proved the higher-dimensional version,
where $m>1$ and $n=1$, in 1990. The general statement we prove here was
conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For
approximations with no coprimality requirement, they also conjectured a
generalized version of Catlin's conjecture, and in 2010 Beresnevich and Velani
proved the $m>1$ cases of that. Catlin's classical conjecture, where $m=n=1$,
follows from the classical Duffin--Schaeffer conjecture. The remaining cases of
the generalized version, where $m=1$ and $n>1$, follow from our main result.
Finally, through the Mass Transference Principle, our main results imply their
Hausdorff measure analogues, which were also conjectured by Beresnevich
\emph{et al} (2009).
