This thesis reports on investigations of a specific collective mode of nuclear vibration, the isoscalar giant monopole resonance (ISGMR), the nuclear "breathing mode", the energy of which is directly related to a fundamental property of nuclei—the nuclear incompressibility. The alpha inelastic scattering experiments reported in this thesis have been critical to answering some fundamental questions about nuclear incompressibility and the symmetry energy, quantities that are crucial to our understanding of a number of phenomena in nuclear physics and astrophysics, including collective excitations in nuclei, radii of neutron stars, and the nature of stellar collapse and supernova explosions. The work described included three sets of experiments and subsequent sophisticated data analysis, both leading to results that have been welcomed by the community and recognised as important contributions to the field.