  Aims: We aim to understand the fragmentation as well as the disk formation,
outflow generation and chemical processes during high-mass star formation on
spatial scales of individual cores.
  Methods: Using the IRAM Northern Extended Millimeter Array (NOEMA) in
combination with the 30m telescope, we have observed in the IRAM large program
CORE the 1.37mm continuum and spectral line emission at high angular resolution
(~0.4'') for a sample of 20 well-known high-mass star-forming regions with
distances below 5.5kpc and luminosities larger than 10^4Lsun.
  Results: We present the overall survey scope, the selected sample, the
observational setup and the main goals of CORE. Scientifically, we concentrate
on the mm continuum emission on scales on the order of 1000AU. We detect strong
mm continuum emission from all regions, mostly due to the emission from cold
dust. The fragmentation properties of the sample are diverse. We see extremes
where some regions are dominated by a single high-mass core whereas others
fragment into as many as 20 cores. A minimum-spanning-tree analysis finds
fragmentation at scales on the order of the thermal Jeans length or smaller
suggesting that turbulent fragmentation is less important than thermal
gravitational fragmentation. The diversity of highly fragmented versus singular
regions can be explained by varying initial density structures and/or different
initial magnetic field strengths.
  Conclusions: The smallest observed separations between cores are found around
the angular resolution limit which indicates that further fragmentation likely
takes place on even smaller spatial scales. The CORE project with its numerous
spectral line detections will address a diverse set of important physical and
chemical questions in the field of high-mass star formation.
