Abstract: We discuss homology and homotopy complexity of manifolds in terms of Gromov’s systolic inequality. The optimal constant in systolic inequality is usually called systolic volume. A central theorem in systolic geometry relates systolic volume to simplicial volume. Since for hyperbolic manifolds there exist proportionality principle, this theorem builds a bridge between systolic geometry and hyperbolic geometry. In the talk, we will present some applications of this theorem to the problem of homology and homotopy complexity of manifolds.