  We introduce the Uchuu suite of large high-resolution cosmological $N$-body
simulations. The largest simulation, named Uchuu, consists of 2.1 trillion
($12800^3$) dark matter particles in a box of side-length 2.0 Gpc/h, with
particle mass $3.27 \times 10^{8}$ Msun/h. The highest resolution simulation,
Shin-Uchuu, consists of 262 billion ($6400^3$) particles in a box of
side-length 140 Mpc/h, with particle mass $8.97 \times 10^{5}$ Msun/h.
Combining these simulations we can follow the evolution of dark matter halos
and subhalos spanning those hosting dwarf galaxies to massive galaxy clusters
across an unprecedented volume. In this first paper, we present basic
statistics, dark matter power spectra, and the halo and subhalo mass functions,
which demonstrate the wide dynamic range and superb statistics of the Uchuu
suite. From an analysis of the evolution of the power spectra we conclude that
our simulations remain accurate from the Baryon Acoustic Oscillation scale down
to the very small. We also provide parameters of a mass-concentration model,
which describes the evolution of halo concentration and reproduces our
simulation data to within 5 per cent for halos with masses spanning nearly
eight orders of magnitude at redshift 0<z<14. There is an upturn in the
mass-concentration relation for the population of all halos and of relaxed
halos at z>0.5, whereas no upturn is detected at z<0.5. We make publicly
available various $N$-body products as part of Uchuu Data Release 1 on the
Skies & Universes site. Future releases will include gravitational lensing maps
and mock galaxy, X-ray cluster, and active galactic nuclei catalogues.
