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The symmetry and structure in momentum-space of the superconducting energy gap inherently depends on the electron-electron pairing interaction responsible for superconductivity. Its experimental determination is therefore imperative in understanding the superconducting state and for comparison with microscopic theories. The temperature dependence of the magnetic penetration depth is a well-established probe of the gap structure as it reflects the energy dependence of the quasiparticle density of states. In this thesis, low temperature tunnel-diode oscillator measurements of the magnetic penetration depth are presented in three low-Tc candidate unconventional superconductors; LaFePO, CsV₃Sb₅ and UTe₂. The magnetic field dependence of the magnetic penetration depth, a manifestation of the nonlinear Meissner effect, is expected to be linear for a nodal superconductor in the clean limit at T = 0. However, this key prediction of unconventional theories of superconductivity remains significantly undertested from an experimental perspective. Presented here are novel field-dependent measurements of the penetration depth in the iron-based superconductor LaFePO, for which a nodal state as been established. The observed linear response to magnetic field at low temperature and further evolution with temperature is consistent with the theory of the nonlinear Meissner effect, further confirming its existence and establishing it as an additional probe of the gap structure. The kagome lattice is predicted to host a number of exotic electronic phases, including quantum spin liquids, chiral charge order, as well as chiral and spin-triplet superconducting states. CsV₃Sb₅ marks one of the first realisations of superconductivity in a compound containing a kagome lattice, however there is still debate over the pairing mechanism and whether the pairing state is nodal or fully-gapped. In this thesis, the temperature dependence of the magnetic penetration depth is presented and reveals a highly anisotropic but fully-gapped superconducting state. This limits the potential pairing mechanisms and favours a conventional electron-phonon mediated pairing scenario. Candidate spin-triplet compound UTe₂ has garnered significant attention due numerous re-entrant superconducting phases that appear in an applied magnetic field. An experimental consensus on the structure of the gap and symmetry of the pairing state in the low-field limit has however not been established, with theoretical studies suggesting a point nodal state aligned along the a-axis. Measurements of the anisotropy of the temperature dependence of the magnetic penetration depth presented here show ubiquitous quadratic behaviour, inconsistent with point-nodes aligned along one of the crystallographic axes in the clean-limit. The ramifications of this result are discussed including the effect of disorder and the possibility of a chiral superconducting state. Finally, the analysis and interpretation of magnetic measurements of superconductors must consider demagnetisation effects on the effective susceptibility of superconducting samples. A finite element analysis study of these effects on spheroidal and cylindrical superconducting geometries is presented and compared to previous numerical studies and analytical approximations.