The QUIJOTE simulations are a set of 44,100 full N-body simulations spanning more than 7000 cosmological models in the {Omega(m), Omega(b), h, n(s), sigma(8), M-nu, w} hyperplane. At a single redshift, the simulations contain more than 8.5 trillion particles over a combined volume of 44,100 (h(-1) Gpc)(3); each simulation follows the evolution of 256(3), 512(3), or 1024(3) particles in a box of 1 h(-1) Gpc length. Billions of dark matter halos and cosmic voids have been identified in the simulations, whose runs required more than 35 million core hours. The QUIJOTE simulations have been designed for two main purposes: (1) to quantify the information content on cosmological observables and (2) to provide enough data to train machine-learning algorithms. In this paper, we describe the simulations and show a few of their applications. We also release the petabyte of data generated, comprising hundreds of thousands of simulation snapshots at multiple redshifts; halo and void catalogs; and millions of summary statistics, such as power spectra, bispectra, correlation functions, marked power spectra, and estimated probability density functions.