  We classify complex projective varieties of dimension $2r \geq 8$ swept out
by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They
are either fibrations onto normal surfaces such that the general fibers are
isomorphic to $\G(1,r)$ or the grassmannian $\mathbb{G}(1,r+1)$. The cases
$r=2$ and $r=3$ are also considered in the more general context of varieties
swept out by codimension two linear spaces or quadrics.
