The concept of a strict deformation quantization provides a mathematical formalism that describes the transition from a classical theory to a quantum theory in terms of deformations of (commutative) Poisson algebras (representing the classical theory) into non-commutative C* -algebras (characterizing the quantum theory). In the first part of this talk we introduce the definitions, give several examples and show how quantization of the closed unit 3-ball $B^3$ in $R^3$ is related to quantization of its smooth boundary (i.e. the two-sphere $S^2$). In the second part we apply this formalism to prove the existence of the 'classical limit' of mean-field quantum spin systems and Schrödinger operators yielding a probability measure on the pertinent phase space. We moreover discuss the concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by taking a limit in the relevant semiclassical parameter.