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 $s_{q^m/q}(k,ρ)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $ρ$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,ρ)$ and evaluate it for certain values of $k$ and $ρ$. We give constructions of rank $ρ$-saturating systems suggested from geometry.