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