Cosmological inference with large galaxy surveys requires theoretical models that combine precise predictions for large-scale structure with robust and flexible galaxy formation modelling throughout a sufficiently large cosmic volume. Here, we introduce the MillenniumTNG (MTNG) project which combines the hydrodynamical galaxy formation model of IllustrisTNG with the large volume of the Millennium simulation. Our largest hydrodynamic simulation, covering (500 Mpc/h)^3 = (740 Mpc)^3, is complemented by a suite of dark-matter-only simulations with up to 4320^3 dark matter particles (a mass resolution of 1.32 x 10^8 Msun/h) using the fixed-and-paired technique to reduce large-scale cosmic variance. The hydro simulation adds 4320^3 gas cells, achieving a baryonic mass resolution of 2 x 10^7 Msun/h. High time-resolution merger trees and direct lightcone outputs facilitate the construction of a new generation of semi-analytic galaxy formation models that can be calibrated against both the hydro simulation and observation, and then applied to even larger volumes - MTNG includes a flagship simulation with 1.1 trillion dark matter particles and massive neutrinos in a volume of (3000 Mpc)^3. In this introductory analysis we carry out convergence tests on basic measures of non-linear clustering such as the matter power spectrum, the halo mass function and halo clustering, and we compare simulation predictions to those from current cosmological emulators. We also use our simulations to study matter and halo statistics, such as halo bias and clustering at the baryonic acoustic oscillation scale. Finally we measure the impact of baryonic physics on the matter and halo distributions.