Formal deformation quantization was introduced as a conceptual approach to the quantization problem and is, by now, well understood: The existence and classification of formal star products can be deduced from Kontsevich's formality theorem. To make the theory more accessible to physics, strict quantization asks to replace the formal parameter by a real number (that plays the role of Planck's constant), in order to obtain a field of well-behaved algebras, such as Fréchet-*-algebras. Quantization is intimately related to symbol calculi for operators and therefore has many applictions, e.g. to index theory.
In this talk, I would like to give a brief introduction to the theory of formal and strict quantization. I will mainly focus on the example of the 2-sphere and explain how to construct an equivariant strict star product on a subclass of analytic functions. Time permitting, I will sketch how to generalize this construction to arbitrary semisimple coadjoint orbits of semisimple connected Lie groups.