Solid-state materials find ubiquitous use in modern technology - from semiconductors in electronics to steel in buildings and superconductors in MRI machines. Theoretical understanding of the atomic-scale behaviour of these materials can be leveraged to design new materials with desirable properties. In this thesis, we investigate the challenges that arise when this is attempted in practice.

Accurate and inexpensive methods to tackle the atomic-scale problem are a prerequisite for materials discovery. We begin with a description of existing methods. This is followed by the development of a Monte Carlo method to calculate expectation values from the many-body picture without the need for a trial wavefunction, which is both a fundamental, and practical, limitation in existing techniques.

Having explored first-principles methods, we turn to their use in understanding materials, beginning with an investigation of the structure of Lithium. Structure searching calculations result in a mixed-phase model at low temperatures, in good agreement with previous experimental and theoretical results. The quasi-harmonic treatment of finite-temperature thermodynamics is extended to include anharmonic nuclear vibrations, which are shown to not alter the phase diagram despite the small mass of the Li atoms.

Focus then shifts towards leveraging these same methods to discover novel superconductors. This begins with an investigation of the LaH$\mathrm{<mrow\; data-mjx-texclass="ORD">10</mrow>}$ and YH$\mathrm{<mrow\; data-mjx-texclass="ORD">10</mrow>}$ compounds, where a new hexagonal phase of LaH$\mathrm{<mrow\; data-mjx-texclass="ORD">10</mrow>}$ provides an explanation for recent experimental measurements. Machine-learning techniques and novel screening methods are then employed to discover hydrides of Rb and Cs that exhibit superconductivity at significantly lower pressures than LaH$\mathrm{<mrow\; data-mjx-texclass="ORD">10</mrow>}$. Optimizations to, and automation of, the workflow then enables the discovery of superconductors on an unprecedented scale, leading to hundreds of new high-temperature superconductors.

Throughout the thesis, the importance of structures that are saddle-points of the energy landscape becomes apparent. The thesis closes with the development of a new algorithm to locate saddle-points that requires no additional information beyond that used by the cheapest existing methods.

This thesis demonstrates that there is progress to be made at every stage of the first-principles materials discovery process and highlights that improving the workflow itself is a non-trivial, but fruitful, pursuit.