We have studied two regions of doubly closed shell nuclei over the nuclear chart: stable nuclei with N ~ Z around 40Ca and highly neutron-rich ones close to drip-line above the 132Sn shell closure. Both experimental and theoretical works have been performed for each of these mass regions. The experimental part includes three works: (i) high energy characterization of the Compton suppressed Clover detector, (ii) In-beam gamma-ray spectroscopic studies of high spin states of 35Cl in mass 40 region and (iii) X- and gamma- ray spectroscopic studies of fission fragments of 252Cf spontaneous fission source in mass 132Sn region. The theoretical study supplemented the experimental findings using the microscopic Shell model and the phenomenological Particle Rotor model. Our work emphasizes that the issues thought to be relevant for nuclei away from stability, should also be studied over a broader region of nuclear territory. With the latest developments in the experimental facilities, the nuclei near the stability line should be re-investigated for finding possible correlations with those away from stability.