  Hydrodynamical simulations of galaxy formation have now reached sufficient
volume to make precision predictions for clustering on cosmologically relevant
scales. Here we use our new IllustrisTNG simulations to study the non-linear
correlation functions and power spectra of baryons, dark matter, galaxies and
haloes over an exceptionally large range of scales. We find that baryonic
effects increase the clustering of dark matter on small scales and damp the
total matter power spectrum on scales up to k ~ 10 h/Mpc by 20%. The non-linear
two-point correlation function of the stellar mass is close to a power-law over
a wide range of scales and approximately invariant in time from very high
redshift to the present. The two-point correlation function of the simulated
galaxies agrees well with SDSS at its mean redshift z ~ 0.1, both as a function
of stellar mass and when split according to galaxy colour, apart from a mild
excess in the clustering of red galaxies in the stellar mass range 10^9-10^10
Msun/h^2. Given this agreement, the TNG simulations can make valuable
theoretical predictions for the clustering bias of different galaxy samples. We
find that the clustering length of the galaxy auto-correlation function depends
strongly on stellar mass and redshift. Its power-law slope gamma is nearly
invariant with stellar mass, but declines from gamma ~ 1.8 at redshift z=0 to
gamma ~ 1.6 at redshift z ~ 1, beyond which the slope steepens again. We detect
significant scale-dependencies in the bias of different observational tracers
of large-scale structure, extending well into the range of the baryonic
acoustic oscillations and causing nominal (yet fortunately correctable) shifts
of the acoustic peaks of around ~5%.
