This thesis presents laser spectroscopy measurements of the hyperfine structures of the neutron-rich indium isotopes 113-131In, which allowed the determination of changes in root-mean-square nuclear charge radii, nuclear spins, magnetic dipole moments and quadrupole moments. These measurements were made at CERN-ISOLDE using the Collinear Resonance Ionization Spectroscopy (CRIS) setup, a high-sensitivity, high-resolution technique. The development of a laser ablation ion source setup allowed the investigation of multi-step laser ionization schemes and measurements of the hyperfine parameters of several atomic states in indium. This lead to the choice of the 246.8 nm (5p 2P3/2 -> 9s 2S1/2) and 246.0 nm (5p 2P1/2 -> 8s 2S1/2) atomic transitions as the most appropriate for the extraction of the nuclear observables. The relative atomic populations of the 5p 2P3/2 and 5p 2P1/2 states were measured using the laser ablation ion source, and were found to be consistent with relative atomic population simulations. The newly measured hyperfine parameters of the 8s 2S1/2 and 9s 2S1/2 states were compared to relativistic coupled-cluster calculations of atomic structure parameters, which gave confidence in the calculated values and enabled the nuclear quadrupole moments in indium to be extracted with unprecedented accuracy. Further comparison to specific mass shift and field shift values extracted from the isotope shift measurements aided in the development of an âanalytic responseâ approach to determine isotope shift constants. This allowed for the first isotone independent determination of the nuclear charge radii of the indium isotopes and of the absolute charge radii for an odd-proton system near the Z = 50 shell closure. The nuclear-model-independent ground and isomeric state electromagnetic properties of indium measured up to N = 82 in this work, particularly of the I=9/2+ and I=1/2- states will refine the understanding of the nuclear structure in this region of the Segre chart. A sudden increase in the I=9/2+ state Âµ values, deviation from the Schmidt value of the I=1/2- state Âµ values and disappearance of odd-even staggering in the ground state mean-square charge radii, among over observations, are yet to be explained by nuclear theory.