We present the ‘Cosmological Jellyfish’ project – a citizen-science classification program to identify jellyfish (JF) galaxies within the IllustrisTNG cosmological simulations. JF are satellite galaxies that exhibit long trailing gas features – ‘tails’ – extending from their stellar body. Their distinctive morphology arises due to ram-pressure stripping (RPS) as they move through the background gaseous medium. Using the TNG50 and TNG100 simulations, we construct a sample of $\sim 80\, 000$ satellite galaxies spanning an unprecedented range of stellar masses, $10^{8.3}{-}10^{12.3}\, \mathrm{M_\odot }$, and host masses, $M_\mathrm{200,c}=10^{10.4}{-}10^{14.6}\, \mathrm{M_\odot }$ back to z = 2. Based on this sample, $\sim 90\, 000$ galaxy images were presented to volunteers in the citizen-science Zooniverse platform, who were asked to determine whether the galaxy image resembles a JF. Based on volunteer votes, each galaxy received a score determining if it is a JF or not. This paper describes the project, the inspected satellite sample, the methodology, and the classification process that resulted in a data set of 5307 visually identified JF galaxies. We find that JF is common in nearly all group- and cluster-sized systems, with the JF fraction increasing with host mass and decreasing with satellite stellar mass. We highlight JF galaxies in three relatively unexplored regimes: low-mass hosts of $M_\mathrm{200,c}\sim 10^{11.5}{-}10^{13}\, \mathrm{M_\odot }$, radial positions within hosts exceeding the virial radius R200, c, and at high redshift up to z = 2. The full data set of our JF scores is publicly available and can be used to select and study JF galaxies in the IllustrisTNG simulations.