Our purpose here is to give an overview of known results and open questions concerning the volume product P(K) = min z∈K vol(K) vol((K − z) * ) of a convex body K in R n . We present a number of upper and lower bounds for P(K), in particular, we discuss the Mahler's conjecture on the lower bound of P(K), which is still open. We also show connections of P(K) with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.