The complex Grassmann manifold G(n,k) of all k-dimensional complex linear subspaces in the complex vector space Cn plays the fundamental role in algebraic topology, algebraic geometry, and other areas of mathematics. The manifolds G(n,1) and G(n,n-1) can be identified with the complex projective space CP(n-1). The coordinate-wise action of the compact torus Tn on Cn induces its canonical action on the manifolds G(n,k). The orbit space CP(n-1)/Tn can be identified with the (n-1)-dimensional simplex. The description of the combinatorial structure and algebraic topology of the orbit space G(n,k)/Tn, where k is not 1 or (n-1), is a well-known topical problem, which is far from being solved. The talk is devoted to the results in this direction which were recently obtained by methods of toric topology jointly with Svjetlana Terzić. The talk is aimed at a broad audience. All necessary definitions and constructions will be given during the lecture.