  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.
