Semiconductor nanostructures exhibit many remarkable electronic and optical properties. The key to designing and utilising semiconductor quantum structures is a physical understanding of the detailed excitation, transport and energy relaxation processes. Thus the nonequilibrium dynamics of semiconductor quantum structures have attracted extensive attention in recent years. Ultrafast spectroscopy has proven to be a versatile and powerful tool for investigating transient phenomena related to the relaxation and transport dynamics in semiconductors. In this thesis, we report investigations into the electronic and optical properties of various semiconductor quantum systems using a variety of ultrafast techniques, including up-conversion photoluminescence, pump-probe, photon echoes and four-wave mixing. The semiconductor quantum systems studied include ZnO/ZnMgO multiple quantum wells with oxygen ion implantation, InGaAs/GaAs self-assembled quantum dots with different doping, InGaAs/InP quantum wells with proton implantation, and silicon quantum dots. The spectra of these semiconductor nanostructures range from the ultraviolet region, through the visible, to the infrared. In the UV region we investigate excitons, biexcitons and oxygen implantation effects in ZnO/ZnMgO multi-quantum wells using four-wave mixing, pump-probe and photoluminescence techniques. Using time-resolved up-conversion photoluminescence, we investigate the relaxation dynamics and state filling effect in InGaAs self-assembled quantum dots with different doping, and the implantation effect in InGaAs/InP quantum wells. Finally, we study the optical properties of silicon quantum dots using time-resolved photoluminescence and photon echo spectroscopy on various time scales, ranging from microseconds to femtoseconds.