The proposal has two parts both of which concern the study of the structure of nuclei under extreme conditions. The first part is the study of nuclei where the proton to neutron ratio is changed to be very different from that found in stable nuclei. At the proton drip line there are so many protons (relative to the number of neutrons) that no more can be added. This is the extreme of stability in terms of the proton neutron ratio. This work will study the properties of nuclei of this type that have a deformed (rugby ball) shape and compare the results to those found in stable nuclei and those predicted by nuclear models. The second part of the work concerns the study of nuclear structure at the extreme of very high spin, just before the nucleus breaks up due to fission. In this work we will concentrate on the role of the pairing force in nuclei near mass 160 at this extreme. The first part of the work will take place at the TRIUMF laboratory in Canada where a new radioactive beam facility and gamma-ray spectrometer become available in 2007. These facilities are vital for this work. For the second part of the work we have already been awarded beam time for an experiment on 160Er and neighbouring nuclei using the GAMMASPHERE spectrometer at the Argonne National Laboratory, USA. This is the only facility worldwide available to carry out this type of detailed nuclear structure work at very high spin. We expect that this experiment will be scheduled in the first half of 2007. The high spin work will be in collaboration with Daresbury Laboratory and will form part of a complementary research programme covering several aspects of the structure of nuclei at the highest spins.