The electronic Hamiltonian contains only pairwise interactions, allowing the energy of an electronic system to be expressed in terms of the two-electron reduced-density-matrix (2-RDM) in lieu of the many-electron wavefunction. The variable space for the exact N-electron wavefunction scales exponentially with the size of the system, while the 2-RDM is polynomial in scale. By using the 2-RDM as the primary variable in electronic structure calculations, it may be possible to obtain very accurate energies at a much more favorable scaling than wavefunction methods. In this thesis, we will use two existing 2-RDM methods to treat electronic systems. First, we will apply the active-space variational 2-RDM method, which directly minimizes the energy with respect to the 2-RDM, to a cadmium telluride polymer that was recently used to greatly enhance the conductivity of CdTe quantum dots. We find that this polymer is very highly correlated despite a deceptively simple structure. We will then turn to the parametric 2-RDM method (p2-RDM), which parameterizes the 2-RDM in terms of a truncated configuration interaction ansatz, but which includes additional flexibility in order to be size-extensive. We apply p2-RDM to the study of the olympicene molecule, which features both fully aromatic and diradical isomers. The parametric 2-RDM method predicts that all isomers are stable to dissociation, in contrast to coupled cluster methods which do not predict stable diradical states. We then present analytical nuclear gradients for p2-RDM, which greatly decrease the number of calculations required to perform geometry optimizations. We apply these gradients to the study of trans-polyacetylene, for which p2-RDM, unlike many wavefunction methods, is able to predict a bond length alternation (BLA) to within experimental values. Lastly, as a single-reference method, p2-RDM may encounter numerical difficulties when the reference wavefunction is of particularly poor quality. We propose a modification to the parameterization that may render the method more generally robust.