Parametrizing subvarieties of a given algebraic variety

Given an algebraic variety $X$ in a projective space, one can try to describe the families of curves (lines, conics...), surfaces (planes, quadrics...), and other subvarieties contained in $X$. These families are themselves algebraic varieties that can help understand the geometry of $X$. I will review the classical case when $X$ is a cubic hypersurface and then continue with the case where $X$ is a so-called Gushel-Mukai variety. This is work in collaboration with Alexander Kuznetsov.