  We introduce the concept of a rank saturating system and outline its
correspondence to a rank-metric code with a given covering radius. We consider
the problem of finding the value of $s_{q^m/q}(k,\rho)$, which is the minimum
$\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank
$\rho$-saturating. This is equivalent to the covering problem in the rank
metric. We obtain upper and lower bounds on $s_{q^m/q}(k,\rho)$ and evaluate it
for certain values of $k$ and $\rho$. We give constructions of rank
$\rho$-saturating systems suggested from geometry.
