We present the Tiered Radio Extragalactic Continuum Simulation (T-RECS): a new simulation of the radio sky in continuum, over the 150 MHz–20 GHz range. T-RECS models two main populations of radio galaxies: Active Galactic Nuclei (AGNs) and Star-Forming Galaxies (SFGs), and corresponding sub-populations. Our model also includes polarized emission over the full frequency range, which has been characterised statistically for each population using the available information. We model the clustering properties in terms of probability distributions of hosting halo masses, and use lightcones extracted from a high-resolution cosmological simulation to determine the positions of haloes. This limits the sky area for the simulations including clustering to a 25 deg2 field of view. We compare luminosity functions, number counts in total intensity and polarization, and clustering properties of our outputs to up-to-date compilations of data and find a very good agreement. We deliver a set of simulated catalogues, as well as the code to produce them, which can be used for simulating observations and predicting results from deep radio surveys with existing and forthcoming radio facilities, such as the Square Kilometre Array (SKA).