We present a detailed study of the hyperfine interaction between carrier and nuclear spins in InAs semiconductor quantum dots. Time resolved measurements on excitons in positively charged quantum dots show the electron spin relaxation due to random fluctuations of the spin orientation of the nuclei in the quantum dot. A complimentary aspect of the hyperfine interaction can be uncovered in single dot continuous wave photoluminescence experiments in a weak magnetic field, namely the Overhauser shift due to the dynamic polarisation of the nuclei following excitation with circularly polarised light.