This paper gives a collection of isoperimetric-type inequalities involving convex bodies, their intersection and projection bodies, polars and Steiner symmetrals, and states some conjectures and open questions. It is shown how some of these inequalities are related to classic isoperimetric problems in Minkowski geometry. A large amount of background material is given, in the form of an overview of classical Euclidean isoperimetric problems, of the isoperimetric problem in finite-dimensional normed spaces, and sections on various notions from Minkowski geometry and convex geometry, including the Holmes-Thompson and Busemann area measures and their isoperimetrices.