In 1932, Cockcroft and Walton performed the alchemist's dream of transforming one chemical element into another when they bombarded lithium with low energy protons from their prototype accelerator, disintegrating it into two helium nuclei. Despite the passage of 70 years from this pioneering work, our understanding of nuclear physics is still largely dictated by what can be achieved by inducing nuclear reactions between stable nuclei i.e. those isotopes which are found in Nature. This has mainly restricted precision studies to nuclei which are close to the line of stability. Our knowledge of nuclear forces and how nuclei behave can only be advanced by studying nuclei with very different numbers of protons and neutrons those of stable isotopes (which are comparatively few in number). A better understanding of the underlying mechanism of nuclear structure is the goal of our research, but it also has consequences beyond nuclear physics. For example it can help our understanding of the processes in supernova explosions where most of the heavy elements found in nature are thought to be synthesised. At the ISOLDE facility, part of the international CERN Laboratory in Geneva, Switzerland, radioactive nuclei are produced with high intensities in the so-called isotope separation on-line (ISOL) technique where a primary target is bombarded with an intense, high energy proton beam. Using different primary targets, ISOLDE can produce beams of varying intensity of over 700 isotopes of 70 different chemical elements. This facility is unique worldwide in the diversity of available beams which it can produce. A recent advance at ISOLDE has been the so-called REX-ISOLDE facility which accelerates these radioactive nuclei to energies where they start to resist the Coulomb repulsion between the positively charge protons when they come into contact with other nuclei in a fixed target. At such energies, interactions take place which allow us to probe the structure of these exotic radioactive nuclei with high precision. Two of these mechanisms are the focus of this grant application. The first, known as Coulomb excitation, is where some of the energy of the interaction goes into exciting the nucleus into higher energy states. The ease with which this takes place reflects the nuclear collectivity, a property which is generally largest for nuclei which are deformed, typically having a non-spherical shape such as a rugby-ball shape, known as prolate deformation. Coulomb-excitation measurements therefore allow us to study the nuclear shape, in particular, a certain special class of nuclei which exhibit shape coexistence. This phenomenon occurs when different states in a particular isotope have different distinct shapes. The second mechanism we aim to employ is known as light-ion transfer. In this interaction between the accelerated beam nucleus and the target nucleus, particles such as protons and neutrons are exchanged. The transferred particle will then occupy one of a number of allowed energy states in the nucleus to which it has been added / there are only a small set of allowed states due to the importance of quantum mechanics in determining the properties of the nucleus. By measuring the ease with which a particle is transferred in such a reaction, we can infer details about the energy states in the nucleus which it has been transferred onto, with high precision. This is especially interesting for the very neutron-rich nuclei since these states are expected to shift around relative to their location in the less exotic nuclei we have studied in the past. It is suggested that this behaviour could very sensitively affect how much of various heavy elements is produced in supernova explosions. Since we know the relative proportions of heavy elements existing in the Solar System, we have a strong constraint on the changes possible in the nuclear physics and a strong motivation for making such studies.