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The quantization of systems with a position dependent mass (PDM) is studied.We present a method that starts with the study of the existence of Killingvector fields for the PDM geodesic motion (Lagrangian with a PDM kinetic termbut without any potential) and the construction of the associated Noethermomenta. Then the method considers, as the appropriate Hilbert space, the spaceof functions that are square integrable with respect to a measure related withthe PDM and, after that, it establishes the quantization, not of the canonicalmomenta $p$, but of the Noether momenta $P$ instead. The quantum Hamiltonian,that depends on the Noether momenta, is obtained as an Hermitian operatordefined on the PDM Hilbert space. In the second part several systems withposition-dependent mass, most of them related with nonlinear oscillators, arequantized by making use of the method proposed in the first part.

Quantization of Hamiltonian systems with a position dependent mass: Killing vector fields and Noether momenta approach