We present an overview of galaxy evolution across cosmic time in the Illustris simulation. Illustris is an N-body/hydrodynamical simulation that evolves 2 × 18203resolution elements in a (106.5Mpc)3box from cosmological initial conditions down to z = 0 using the AREPO moving-mesh code. The simulation uses a state-of-the-art set of physical models for galaxy formation that was tuned to reproduce the z = 0 stellar mass function and the history of the cosmic star formation rate density. We find that Illustris successfully reproduces a plethora of observations of galaxy populations at various redshifts, for which no tuning was performed, and provide predictions for future observations. In particular, we discuss (a) the buildup of galactic mass, showing stellar mass functions and the relations between stellar mass and halo mass from z = 7 to 0, (b) galaxy number density profiles around massive central galaxies out to z = 4, (c) the gas and total baryon content of both galaxies and their haloes for different redshifts, and as a function of mass and radius, and (d) the evolution of galaxy specific star formation rates up to z = 8. In addition, we (i) present a qualitative analysis of galaxy morphologies from z = 5 to 0, for the stellar as well as the gaseous components, and their appearance in Hubble Space Telescope mock observations, (ii) follow galaxies selected at z = 2 to their z = 0 descendants, and quantify their growth and merger histories, and (iii) track massive z = 0 galaxies to high redshift and study their joint evolution in star formation activity and compactness. We conclude with a discussion of several disagreements with observations, and lay out possible directions for future research.