In this thesis , ZnO nanoparticles with different cobalt concentration were prepared by a simple and rapid method . This method is based on a short time solid state milling and calcinations of zinc acetate , cobalt acetate , and citric acid powders . The physical properties of the samples were characterized using X-ray diffraction (XRD) , Field Emission Scanning Electron Microscopy (FE-SEM) , Fourier Transform Infrared (FTIR) , photoluminescence , and UV-vis . spectroscopy . It was shown that a very low substitution of Co (less than 10 percent of molecular weight) has little effect on the lattice parameters of ZnO and significantly decreases the band gap (Eg ) value of the synthesized ZnO:Co nanoparticles . Calculation based on the XRD data shows that the average crystallite sizes of ZnO particles are nearly 18 nm . Photoluminescence spectroscopy shows that many defects such as interstitial zinc , zinc vacancy , and exciton recombination are responsible for the observed optical properties . Magnetization measurements which were recorded by a superconducting quantum interference device (SQUID) magnetometer determine the paramagnetic behaviour for all samples due to the absence of enough oxygen vacancy . To study the nature of magnetic properties of this system , Density functional theory was employed . DFT can not predict correct band gap in semiconductor due to self interaction error . so for predicting correct band gap and properties related that the many body GW approximation and ASIC approach was used . First single shot GW (G0W0) and self consistent GW (scGW) quasiparticle density of states upon GGA , GGA+U and HSE03 for Co doped zinc oxide are investigated . The wurtzite crystal field split minority 3d-Co states to e and t 2 , which the location of them is in doubt , especially with LDA and GGA calculation . In consideration to new many body scenario (GW) and its reputation to predict E g and d binding energy in wide band semiconductor such as ZnO , ZnS and LiF , we have predicted the position of t 2 to be equal to 1.4-4.3~eV above the conduction band minimum i.e . it is impossible to predict the presence of ferromagnetism in the absence of defects in zinc oxide doped with cobalt . Then , In order to better understand the role of extended defect structures in stabilizing long range magnetic order in this material , the electronic structure of transition metal Co doped ZnO grain boundaries (GBs) was studied . Recent experimental work has shown that grain boundaries and defects associated to them possibly play a fundamental role in the observed ferromagnetism of Co doped ZnO . This observed correlation between the presence of GBs and ferromagnetism will be investigated in this work with the help of ab-initio electronic structure calculations . Identifying qualitative as well as quantitative differences between the electronic structures of Co dopant in the bulk material and at the grain boundary is key to understand this phenomenon . Then , we shall be employing self-interaction corrected density functional theory (DFT) methods incorporated into the SIESTA platform . And at the end , the density functional - full potential computations are performed to study the ring to cage structural crossover in the small (ZnO)n (n=2-16) clusters and its effects on electronic and vibrational thermodynamic behaviour of the system . The origin of this structural crossover , which is found to occurs at n = 10 , is studied by the investigation of the behavior of the Zn-O-Zn bond angle , the Zn-O bond strength , and the number of bonds in the systems . It is found that 12 is the lowest magic number of ZnO clusters at ground state , while finite temperature vibrational excitations enhance the relative stability of the (ZnO) 9 cluster and make it a magic system at temperatures above about 170 K . The obtained electronic structure of ZnO clusters before and after applying the many-body GW corrections evidence a size induced red shift originated from the ring to cage structural crossover in the system . The behaviour of the extremal points of electron density of the clusters along with the extrapolated cluster binding energies at very large sizes may be evidences for existence of a metastable structure for large ZnO nanostructures , different with the bulk ZnO structure . Keywords: ZnO:Co , Many-body GW approximation , Atomic Self Interaction Correction (ASIC) , ZnO Clusters .