A systematic investigation of the nuclear observables related to the triaxial degree of freedom is presented using the multi-quasiparticle triaxial projected shell model (TPSM) approach. These properties correspond to the observation of $gamma$-bands, chiral doublet bands and the wobbling mode. In the TPSM approach, $gamma$-bands are built on each quasiparticle configuration and it is demonstrated that some observations in high-spin spectroscopy that have remained unresolved for quite some time could be explained by considering $gamma$-bands based on two-quasiparticle configurations. It is shown in some Ce-, Nd- and Ge-isotopes that the two observed aligned or s-bands originate from the same intrinsic configuration with one of them as the $gamma$-band based on a two-quasiparticle configuration. In the present work, we have also performed a detailed study of $gamma$-bands observed up to the highest spin in Dysposium, Hafnium, Mercury and Uranium isotopes. Furthermore, several measurements related to chiral symmetry breaking and wobbling motion have been reported recently. These phenomena, which are possible only for triaxial nuclei, have been investigated using the TPSM approach. It is shown that doublet bands observed in lighter odd-odd Cs-isotopes can be considered as candidates for chiral symmetry breaking. Transverse wobbling motion recently observed in $^{135}$Pr has also been investigated and it is shown that TPSM approach provides a reasonable description of the measured properties.
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