High energy density (HED) matter is defined as having an energy density above $10^{11}$ $J•m^{-3}$, which is equivalent to 100 GPa (1 Mbar) pressure and 500 T magnetic pressure. The scientific objectives of the High Energy Density Physics instrument (HED instrument) at the European XFEL are to create and explore HED states using the particular features of X-ray free-electron laser (FEL) radiation. Strongly excited solid but below-HED states are also considered to be a key science area, as such transient states are precursors to the formation of HED states in laboratory experiments. This report describes the conceptual design of the HED instrument at the European XFEL. The HED instrument will be one of the six baseline instruments of the facility. It will provide to a wide community of condensed-matter, plasma, and high-power laser physicists a scientific instrument for experiments not possible anywhere else in the world.This report is based on the requirements emerging from the scientific objectives of the HED instrument and the anticipated X-ray techniques to be fielded at this instrument. Major subsystems of the HED instrument are described, including the beam transport elements, the instrument components, and the requirements for further subsystems and infrastructure. Furthermore, the interfaces of the HED instrument to other work packages of the European XFEL project are described with the goal of realizing a highly integrated project.