Geometric variational problems are characterized by scale invariance and therefore often lack the compactness properties required for the use of variational methods. The discovery of threshold phenomena and, in fact, quantization of the energy levels where compactness fails lead to the resolution of a number of classical conjectures in Geomeric Analysis, such as Rellich's conjecture or convergence of the Yamabe flow. In my talk I will survey some of these results and conclude with recent quantization results for the class of elliptic equations related to the Moser-Trudinger-Adams inequality in a "super-critical" regime, whose concentration behavior is governed by a geometric "limit equation".