In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

Comment: Accepted for publication in Communications in Mathematics

• 11H06 - Lattices and convex bodies (number-theoretic aspects)
• 52A39 - Mixed volumes and related topics in convex geometry
• 52A40 - Inequalities and extremum problems involving convexity in convex geometry
• 52C07 - Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)