Starting with Minkowski inequalities bounding the volume of a convex body in terms of its successive minima, we survey on some classical and recent results (and conjectures) regarding similar relations where either i) the volume functional is replaced (e.g., by the number of lattice points of the body or by the volume of the polar body) or/and ii) the successive minima are replaced by another series of functionals. In particular in ii) we are interested in the so called packing minima, where the i-th packing minimum of a convex body K is the maximal length of a shortest lattice vector among all projections of the given lattice along (n-i)-dimensional lattice planes, and the length is measured here with respect to the projection of K-K.