In this thesis, I use X-ray total scattering methods, neutron powder diffraction and ab initio methods to study the structures of nanoparticles. The main material I study is CdS magic-size clusters (MSCs). It has been long noted that colloidal QDs prepared within the same batch could be readily divided into regular quantum dots (RQDs) and magic-sized clusters (MSCs) with the size range of 1-2 nm. Since the first discovery of QDs in 1983, QDs have wide applications in light-emitting diodes, biological labelling, solar cell etc. Total scattering as a powerful technique has been widely used in structure determinations in materials for many years. However, the MSCs studied here is more complicated. It is more challenging to get structure information for MSCs from conventional methods because of its small size. This work aims to give some insights and inspirations in this field of studying the atomic structures of MSCs. The results showed that RQDs have structures similar to those of the two bulk crystalline phases with random stacking of layers. The MSCs have a similar local structure of corner-linked (SCd)4tetrahedra but with different longer-range connectivities. I also combined simulation methods in order to get more structure model candidates. I carried out a study of (CdS)n and (CdSe)n nanoclusters, with n up to 34, using the ab initio random structure searching method. Many stable nanoclusters were discovered. With increasing n the clusters showed a gradual transformation from cage-like to bulk-like structures. Strikingly, for any value of n, the range of structures had a continuous distribution of cluster energies. In the journey of study the atomic structure of MSCs, I found CdS MSCs has a special property that the UV-vis absorption peak changed back and forth with the temperature increase and decrease. Thus a study about observing solid- 5 6 state structure transformation at an atomic level using in situ X-ray total scattering was developed. I showed that the transitions are affected by the synthesis method and surface ligands. In order to learn more about another experimental technique - neutron diffraction. I performed a neutron diffraction experiment of Fe3O4 nanoparticles under pressure, which showed that the nanoparticles become softer with decreasing particle size. Separately I calculated the structures and lattice dynamics of crystalline malononitrile using density functional theory methods. The results confirmed the proposed structure of the δ phase, showing good agreement between calculated and experimental structures and spectroscopic frequencies. The calculations showed the soft mode associated with the α-β-γ displacive phase transitions, and that the entropy associated with the α-β-γ phase transitions gives the stability of these phases over that of the phase at high temperatures.