Charge ordering is a phenomenon in which the electron field in a material spontaneously breaks the symmetry of the underlying crystal lattice, selecting out a new period and, in dimensions higher than one, a particular direction in space. In the first half of this thesis I study the consequences of charge ordering in ID, applying a self-consistent mean-field approach. A system with rational filling p/q forms a period q charge density wave, which I demonstrate exhibits a quantized adiabatic particle transport upon being dragged through a full period. I show that an irrationally-filled system is quasiperiodic, and use the equivalence to show that ID quasicrystals fit into the topological classification of free fermion systems known as the Tenfold Way. Using a free energy analysis I demonstrate that incommensurate charge order provides a new non-local growth mechanism for ID quasicrystals, potentially greatly increasing the number of known, naturally-occurring, examples. In the second half of this thesis I address the question of whether the ID charge ordering mechanism, the Peierls instability, applies in dimensions higher than one, focussing on the prototypical 2D charge-ordered material niobium diselenide, NbSe2' In this case I definitively rule out such 'weak-coupling' theories, and show that it is necessary to consider a model of a strong electron-phonon coupling dependent on both the ingoing and outgoing electron momenta and the electronic bands scattered between. The model provides the first consistent theoretical account of the full range of experimental results on the system, including a particle/hole asymmetric gap centred above the Fermi level which opens in one band only, the softening of phonon frequencies over a wide range of momenta, and the existence of a pseudogap regime over a range of temperatures, with the latter explained as suppression of charge order through fluctuations of the phonon field.