Some of the most exciting contemporary problems in condensed matter physics are in the area of nanoscale quantum mechanics and relate to the properties of superconducting and ferromagnetic materials at ultra-low temperatures, e.g. in the study of macroscopic quantum tunnelling and novel forms of quantum order. A key aspect of this work is the ability to perform high spatial resolution magnetic imaging down to ultra-low temperatures and/or at high magnetic fields. Due to the major technical challenges posed in these regimes, no magnetic imaging system currently exists which is capable of nanoscale imaging. Over the last decade we have successfully developed a number of scanning Hall probe microscopes (SHPMs) in Bath which span the temperature range 4.2-300K. These Hall probe systems represent a very flexible approach to magnetic imaging since Hall sensors can readily be fabricated with ~100nm spatial resolution and excellent minimum detectable fields and, using high probe current densities, can be operated in magnetic fields of several Teslas. All existing SHPM systems use 4He exchange gas cooling and, hence, can only be operated above 4.2K. In order to address a range of exciting contemporary problems in magnetic materials at much lower temperatures we propose to design and construct a new SHPM head which can be directly attached to the 300mK cold pot of a commercial 3He cryostat insert. This instrument will be capable of operation in magnetic fields up to 10T with a spatial resolution >100nm and minimum detectable field ~10mG/Hz^0.5. Its unique properties will be exploited to open up a new front in the field of nanoscale quantum phenomena. Within the project we will search for novel vortex structures and spontaneously generated flux in the unconventional superconductor Sr2RuO4, and will attempt to establish the mechanism for macroscopic quantum tunnelling of the magnetisation in the molecular magnet Mn12-acetate. In addition we will search for new geometry-driven vortex structures in mesoscopic Al disks.