This thesis reports the structural and magnetic behaviour of complex lanthanide oxides exhibiting low-dimensional and/or geometrically frustrated magnetism. Materials displaying these phenomena are well studied as hosts for novel magnetic states such as spin liquids, and for applications such as magnetic refrigeration. In this thesis, three families of materials were studied in order to probe relationships between their structural and magnetic properties. First, the lanthanide calcium oxyborates Ca4LnO(BO3)3, Ln = Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Yb, were synthesised using solid-state ceramic methods. In Ca4LnO(BO3)3 the magnetic Ln3+ ions form quasi-one-dimensional chains. The crystal structure was investigated using powder X-ray and neutron diffraction and the bulk magnetic properties were determined by magnetic susceptibility and heat capacity. There is evidence for short-range ordering in Ca4TbO(BO3)3 at 3.6 K but no long- or short-range order above 1.8 K in the other compounds. The magnetocaloric performance of Ca4GdO(BO3)3 and Ca4HoO(BO3)3 was found to be comparable with the well-known magnetic refrigerants Gd3Ga5O12 and Dy3Ga5O12 respectively. Secondly, the structural behaviour of the barium borates Ba3Ln(BO3)3 was probed as a function of temperature and lanthanide ion. A new low-temperature phase of Ba3Tb(BO3)3 was observed with the hexagonal crystal structure previously known only for Ln = Dy–Lu. In this phase the magnetic Tb3+ ions form quasi-two-dimensional triangular layers. The new phase was examined using powder diffraction and magnetometry and found to be strongly geometrically frustrated: low-temperature neutron diffraction indicates the absence of long-range magnetic order above 75 mK. Muon spectroscopy above 1.5 K also shows an absence of long-range order. Additional experiments are required for further investigation into the possible quantum spin liquid ground state. Finally, bulk magnetic properties are reported for the lanthanide orthotantalates LnTaO4, Ln = Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er. The lanthanide ions in LnTaO4 form an elongated diamond-like lattice which can theoretically host unusual magnetic states in the presence of competing magnetic interactions. Neutron diffraction revealed long-range antiferromagnetism in TbTaO4 while the remaining compounds do not order above 1.8 K. This thesis reports previously unexplored fundamental magnetic properties of several different lanthanide materials, demonstrating the rich variety of magnetic behaviour accessible via chemical substitution of the lanthanide ions in solid-state compounds.