The linear optical properties and exciton dynamics of different semiconductor nanostructures have been studied. A model to describe the absoprtion spectra of cadmium selenide (CdSe) nanoplatelets (NPLs) has been developed, which allows the extraction of parameters relating to excitons confined within the thickness of the NPLs. Giant oscillator strength transitions (GOST) have been observed in NPLs with a lifetime limited dephasing of the ground state excitons at low temperature of about 1 ps, using transient resonant four wave mixing in heterodyne detection. The observation of the GOST effect has been affirmed by a decrease in the low temperature dephasing time with increasing NPL area. In addition, in cadmium selenide/cadmium sulphide (CdSe/CdS) quantum dot in rod (QDR) samples, dephasing dynamics at low temperature have been described by a Gaussian distribution of decay rates. Density dynamics have shown a fast initial decay followed by a signal rise attributed to an interplay between acoustic phonons and fast population dynamics related to the QDR fine structure. Density dynamics at longer delays have been attributed to an ensemble including both excitons and trions. The low temperature dephasing has been attributed to relaxation of excitons from a bright state to a lower lying dark state. Finally, a homebuilt scanning probe microscopy instrument has been modified to allow simultaneous optical and scanning probe imaging for the study of correlation between charge state and optical properties of nanostructures.