We review the current knowledge about nuclear star clusters (NSCs), and the spectacularly dense and massive assemblies of stars found at the centers of most galaxies. Recent observational and theoretical works suggest that many NSC properties, including their masses, densities, and stellar populations, vary with the properties of their host galaxies. Understanding the formation, growth, and ultimate fate of NSCs, therefore, is crucial for a complete picture of galaxy evolution. Throughout the review, we attempt to combine and distill the available evidence into a coherent picture of NSC evolution. Combined, this evidence points to a clear transition mass in galaxies of ∼109M⊙ where the characteristics of nuclear star clusters change. We argue that at lower masses, NSCs are formed primarily from globular clusters that inspiral into the center of the galaxy, while at higher masses, star formation within the nucleus forms the bulk of the NSC. We also discuss the co-existence of NSCs and central black holes, and how their growth may be linked. The extreme densities of NSCs and their interaction with massive black holes lead to a wide range of unique phenomena including tidal disruption and gravitational-wave events. Finally, we review the evidence that many NSCs end up in the halos of massive galaxies stripped of the stars that surrounded them, thus providing valuable tracers of the galaxies' accretion histories.