Mathematical Foundations of Geometric Quantization

In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian connections, real and complex polarizations, metalinear bundles and bundles of densities and half-forms. In addition, we justify all the steps followed in the geometric quantization programme, from the standpoint definition to the structures which are successively introduced.