This thesis presents significant new observations of nuclear wobbling, and thus expands our understanding of nuclear triaxiality and its prevalence in the nuclear chart. Triaxial nuclear shapes are a very rare phenomena and their experimental identification often relies on two unique signatures – nuclear wobbling motion and chiral rotation. While nuclear chirality is a well-studied phenomenon, experimental observations of wobbling nuclei are rather limited. With the identification of ¹³⁵Pr and ¹⁸⁷Au as wobblers, this work establishes triaxiality to be a general phenomenon present in different regions of the nuclear chart, irrespective of any particular spin or deformation. A major focus of this work is the detailed investigation of the different kinds of wobbling modes. Depending on the geometry of the nuclear system, wobbling can be classified into two types – longitudinal and transverse. This work has, for the first time, reported evidence of the possible coexistence of both forms of wobbling in a single nucleus. Another important result reported in this work is the very first observation of co-existing chiral and wobbling modes in the ¹³⁵Pr nucleus. This thesis details the experimental methods that led to this breakthrough, along with pertinent theoretical interpretations.