The polar formalism is a mathematical method, based upon a complex variable transformation, proposed in 1979 by G. Verchery for representing plane tensors of any rank using invariants and angles. As such, it is particularly suited for representing anisotropic properties, in particular elasticity. In this paper, we give a brief account of the fundamentals of the polar formalism, stressing in particular the role played by the polar invariants on the characterization of elastic symmetries, that leads to a new classification of them, based upon an algebraic criterion and that has allowed for the discovery of two special orthotropies. Then, we focus on some special theoretical subjects: anisotropy of complex or rari-constant layers, some strange cases of interaction between geometry and anisotropy, the anisotropy of damaged layers initially isotropic.