The geometrical and topological aspects of quantum field theory and gravity are covered in this monograph . It is well known that chiral anomaly, which is related to the persistence of axial vector current divergence, emerges from the interaction of a chiral fermion with a gauge field. The insight that this is tied to particular topological properties associated with the fermion leads to the realisation of the topological origin of the fermion number and the Berry phase. The participation of gauge fields in the quantization process has an impact on a fermion's topological properties, allowing...