We examined Zeeman splitting in a series of InₓGa₍₁₋ₓ₎As/GaAs self assembled quantum dots (SAQD's) with different pump polarisations. All these measurements were made in very low external magnetic fields where direct determination of the Zeeman splitting energy is impossible due to it's small value in comparison to the photoluminescence linewidths. The use of technique developed by M. J. Snelling allowed us to obtain the Zeeman splitting and hence the excitonic g-factors indirectly. We observed a linear low field splitting, become increasingly non-linear at higher fields. We attribute this non-linearity to field induced level mixing. It is believed these are the first low field measurements in these structures. A number of apparent nuclear effects in the Zeeman splitting measurements led us onto the examination of nuclear effects in these structures. The transverse and oblique Hanle effects then allowed us to obtain the sign of the electronic g-factors in two of our samples, for one sample, a (311) gown In₀.₅As/GaAs SAQD sample, we were able to ascertain the spin relaxation time, the maximum value of the nuclear field, and provide evidence of the existence of nuclear spin freezing in at least one of our samples. We have then used a novel technique investigated by D. J. Guerrier, to examine optically detected nuclear magnetic resonance in our samples. We believe this is the first such study on these structures. We could not ascertain the dipolar indium resonance signal, even though all other isotopes were seen. We have therefore suggested a number of possible mechanisms that may be responsible for the lack of an indium resonance signal.