  The aim of this note is to survey the results in some geometric problems
related to the centroids and the static equilibrium points of convex bodies. In
particular, we collect results related to Gr\"unbaum's inequality and the
Busemann-Petty centroid inequality, describe classifications of convex bodies
based on equilibrium points, and investigate the location and structure of
equilibrium points, their number with respect to a general reference point as
well as the static equilibrium properties of convex polyhedra.
