One important area of modern condensed matter research is the investigation of the nature of the superconducting cuprates. Much progress in this field has been obtained with the technique of neutron scattering. We here present a review of neutron scattering studies of the high-temperature superconductor La2-xSrx CuO4, performed at Riso National Laboratory. We review the work on the mapping of the incommensurate spin fluctuations, the investigation of the gap in the fluctuation spectrum, the magnetic properties of the vortices appearing in an applied field, and the quantum critical behaviour of the system. We discuss our findings in the light of results of neutron scattering from other groups, on other cuprate systems, and results from other experimental methods, e.g. NMR, muSR, STM, X-ray diffraction, and ARPES. We end with a discussion on the implications of the experimental results for the progress in the general understanding of high-temperature superconductivity.