The spin degree of freedom of charge carrier spins and the host’snuclear spins in semiconductors are potential sources for the nextgeneration of spintronics applications which motivate the deliberateinvestigation of the spin dynamics of well-controlled model systemslike n-GaAs. Conduction electron spins are mobile in semiconductors and can be initialized, manipulated, and read out optically.Optical pumping with circularly polarized light can for examplecreate a non-equilibrium electron spin polarization close to 100%.Nuclear spins are practically appealing as well due to their very longspin relaxation times. The mutual interaction between the electronand nuclear spin system is mediated via the hyperfine interaction.Indeed, through this interaction, a non-equilibrium spin polarization of electrons is transferred to the nuclear spins and results indynamic nuclear polarization, which inter alia has an intricate dependence on the doping density. The main objectives of this thesisare measuring most accurately (i) the temperature dependence ofthe electron spin relaxation rate and (ii) the magnetic field, doping,and temperature dependence of the nuclear spin relaxation rate ina set of high quality n-GaAs samples from quasi-insulating over themetal-to-insulator transition up to the quasimetallic regime.The temperature dependence of the electron spin relaxation timeis measured very accurately for three of the above-mentioned samples with the optical Hanle depolarization method. The measurements yield, in combination with a theoretical model, a quantitativeinsight into the efficiency of the different spin relaxation mechanisms. The longest electron spin relaxation time in n-GaAs resultsfrom an interplay of variable range hopping and hyperfine interaction for a doping concentration just below the Mott metal-to insulator transition at a finite temperature of ∼ 7 K. At higher doping densities the effect of these two mechanisms decreases suchthat they are negligible in the highest doped sample. For moderateand high temperatures, the description of the electron spin relaxation becomes unpretentious since the Dyakonov-Perel mechanismdominates over all other electron spin relaxation mechanisms.The Overhauser field from nuclear polarization intensifies or weakens the external magnetic field and affects the electron spin orientation. In order to measure the nuclear spin relaxation, a three-stagetime-resolved detection of the Hanle effect is used. The method includes optical pumping and measuring the difference of the nuclearspin polarization before and after a dark (no laser light) intervalof variable length. In this way, the nuclear spin system in the absence of excitation is investigated. The magnetic field dependenceof the nuclear spin relaxation rate has a typical Lorentzian shape,showing the spin-spin interaction’s impact at lower magnetic fields.The strong field doping dependence of the nuclear spin relaxationrate can be explained quantitatively, considering the effective number of localized electrons over the entire density regime. Nuclearspin diffusion to the donor bound electrons increases the relaxationrate of the nuclear spin measured at 6:5 K and results in a distinctmaximum at the metal-to-insulator transition. The rate in the veryhigh doped sample increases due to the Korringa mechanism. Theinvolved mechanisms explain the trend of the relaxation except forthe very low doped sample. The temperature dependence of thelowest doped sample shows an electron spin relaxation channel affecting the nuclear spin relaxation, which is negligible at high dopingdensities.