Future observations of cosmic microwave background (CMB) polarisation have the potential to answer some of the most fundamental questions of modern physics and cosmology, including: what physical process gave birth to the Universe we see today? What are the dark matter and dark energy that seem to constitute 95% of the energy density of the Universe? Do we need extensions to the standard model of particle physics and fundamental interactions? Is the ACDM cosmological scenario correct, or are we missing an essential piece of the puzzle? In this paper, we list the requirements for a future CMB polarisation survey addressing these scientific objectives, and discuss the design drivers of the CORE space mission proposed to ESA in answer to the "M5" call for a medium-sized mission. The rationale and options, and the methodologies used to assess the mission's performance, are of interest to other future CMB mission design studies. CORE has 19 frequency channels, distributed over a broad frequency range, spanning the 60-600 GHz interval, to control astrophysical foreground emission. The angular resolution ranges from 2' to 18', and the aggregate CMB sensitivity is about 2 mu K.arcmin. The observations are made with a single integrated focal-plane instrument, consisting of an array of 2100 cryogenically-cooled, linearly-polarised detectors at the focus of a 1.2-m aperture cross-Dragone telescope. The mission is designed to minimise all sources of systematic effects, which must be controlled so that no more than 10(-4) of the intensity leaks into polarisation maps, and no more than about 1% of E-type polarisation leaks into B-type modes. CORE observes the sky from a large Lissajous orbit around the Sun-Earth L2 point on an orbit that offers stable observing conditions and avoids contamination from sidelobe pick-up of stray radiation originating from the Sun, Earth, and Moon. The entire sky is observed repeatedly during four years of continuous scanning, with a combination of three rotations of the spacecraft over different timescales. With about 50% of the sky covered every few days, this scan strategy provides the mitigation of systematic effects and the internal redundancy that are needed to convincingly extract the primordial B-mode signal on large angular scales, and check with adequate sensitivity the consistency of the observations in several independent data subsets. CORE is designed as a "near-ultimate" CMB polarisation mission which, for optimal complementarity with ground-based observations, will perform the observations that are known to be essential to CMB polarisation science and cannot be obtained by any other means than a dedicated space mission. It will provide well-characterised, highly-redundant multi-frequency observations of polarisation at all the scales where foreground emission and cosmic variance dominate the final uncertainty for obtaining precision CMB science, as well as 2' angular resolution maps of high-frequency foreground emission in the 300-600 GHz frequency range, essential for complementarity with future ground-based observations with large telescopes that can observe the CMB with the same beamsize.