The Earth extends some 6400 km to the centre where the conditions of pressure (P) and temperature (T) reach over three million times atmospheric pressure and ca.6000°C. We stand on thin brittle crustal plates moving through geological time over a continuously deforming mantle of slowly convecting hot rock. The mantle extends about halfway through the Earth to a liquid outer core and a solid inner core. Although the mantle and core make up 99% of the Earth by volume and mass, we are only able to sample mantle material directly to a few hundred kilometres, from inclusions in diamonds that are brought up to the surface by volcanic intrusions; the remaining 90% of the Earth is effectively inaccessible. The most direct knowledge we have of the Earth's deep interior comes from the seismic waves generated from earthquakes. A knowledge of material properties coupled with these seismic waves tell us that the mantle is made up of complex silicates and that the core is predominantly made of solid and liquid iron with some alloying elements. However, the detailed structure of the Earth's deep interior is poorly constrained. Major advances toward the understanding of the composition, structure and dynamics of the Earth's deep interior are to be gained only by a combination of experimental and theoretical techniques. It is already clear that many of the large–scale geological processes responsible for the conditions at the surface are driven from the Earth's core. However, there are many questions yet to be answered about the exact nature of the core and mantle, and the interaction between them. For example, we have yet to fully define the major– and minor–element chemistry of the mantle, the convective regime of the mantle, the alloying elements in the core, the nature of the core–mantle boundary and the dynamical processes in the outer core governing the geodynamo. Advances in high–P/T experimental techniques over the last two decades allow laboratory simulation of the physical conditions from the surface of the Earth to the core, shedding light on the physics and chemistry of the Earth's deep interior. High P and T can be maintained for significant periods (minutes to days) in multi–anvil and diamond–anvil presses. Shock experiments produce high T and P in the megabar range for tiny durations (milliseconds), but, in doing so, they shed light on the physics of the solid inner core. The current development of in situ high–pressure research such as P– and S–wave interferometry, electrical conductivity and synchrotron–based X–ray techniques will, over the coming decades, allow significant improvements in our understanding of processes in the deep Earth. Even so, with increasing depth, it becomes increasingly difficult to mimic the extreme conditions of P and T precisely. An alternative to laboratory experiments is the use of computer simulations, which allow us to test which models best match the seismic evidence and experimental data. In particular, with increasingly powerful supercomputer resources, emphasis is now being placed on the use of ab initio quantum–mechanical calculations to simulate materials at the conditions of pressure and temperature to be found in the Earth's deep interior. This approach allows us to predict the properties of candidate mantle silicates with remarkable accuracy when compared with seismic data and the results of laboratory experiments. With these simulation techniques, we are also trying to solve many problems that are out of the reach of experimentation involving simultaneously high P and T, such as the nature of iron and iron alloys under the extreme conditions of the core where iron is squeezed to about half its normal volume, and we will soon be able provide constraints on the temperature profile of the Earth, which, at core depths, is known only to within a few thousand degrees! It is therefore the challenge of the next few years among deep–Earth scientists to develop accurate measurements and models of the properties of the high–pressure silicates and iron alloys at deep–Earth conditions. With an interdisciplinary approach involving theory, experiment and seismology we will be able to determine the nature, evolution and influence of the Earth's deep interior.