Classical Schubert calculus deals with orbits of a Borel subgroup in $GL(V)$ acting on a Grassmann variety $Gr(k,V)$ of $k$-planes in a finite-dimensional vector space $V$. These orbits (Schubert cells) and their closures (Schubert varieties) are very well studied both from the combinatorial and the geometric points of view. One can go one step farther, considering the direct product of two Grassmannians and a Borel subgroup in $GL(V)$ in acting diagonally on this variety. In this case, the number of orbits still remains finite, but their combinatorics and geometry of their closures become much more involved. However, something still can be said about them. I will explain how to index the closures of a Borel subgroup in $Gr(k,V)\times Gr(l,V)$ combinatorially and construct the resolutions of their singularities, which are similar to Bott-Samelson resolutions for ordinary Schubert varieties. I will also speak about the analogues of these results for direct products of partial flag varieties for reductive groups of types different from $A_n$, due to P. Achinger, N. Perrin, and myself.