Francesco Maggi, Research Scientist at ICTP, works on variational problems and partial differential equations motivated by geometry and physics. He is contributing, in particular, to the development of a quantitative stability theory for geometric/functional inequalities, and for geometric rigidity theorems, as well as to the applications of this theory to minimization problems arising in capillarity theory and statistical mechanics, and to the asymptotic behavior of geometric flows and other diffusive partial differential equations. Abstract: I will review several isoperimetric theorems, pointing out some (more or less well-known) open problems. I then introduce some key results in the theory of quantitative isoperimetry. The necessary mathematical background should be minimal.