  We forecast the main cosmological parameter constraints achievable with the
CORE space mission which is dedicated to mapping the polarisation of the Cosmic
Microwave Background (CMB). CORE was recently submitted in response to ESA's
fifth call for medium-sized mission proposals (M5). Here we report the results
from our pre-submission study of the impact of various instrumental options, in
particular the telescope size and sensitivity level, and review the great,
transformative potential of the mission as proposed. Specifically, we assess
the impact on a broad range of fundamental parameters of our Universe as a
function of the expected CMB characteristics, with other papers in the series
focusing on controlling astrophysical and instrumental residual systematics. In
this paper, we assume that only a few central CORE frequency channels are
usable for our purpose, all others being devoted to the cleaning of
astrophysical contaminants. On the theoretical side, we assume LCDM as our
general framework and quantify the improvement provided by CORE over the
current constraints from the Planck 2015 release. We also study the joint
sensitivity of CORE and of future Baryon Acoustic Oscillation and Large Scale
Structure experiments like DESI and Euclid. Specific constraints on the physics
of inflation are presented in another paper of the series. In addition to the
six parameters of the base LCDM, which describe the matter content of a
spatially flat universe with adiabatic and scalar primordial fluctuations from
inflation, we derive the precision achievable on parameters like those
describing curvature, neutrino physics, extra light relics, primordial helium
abundance, dark matter annihilation, recombination physics, variation of
fundamental constants, dark energy, modified gravity, reionization and cosmic
birefringence. (ABRIDGED)

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory