The formation of globular clusters (GCs) remains one of the main unsolved problems in star and galaxy formation. The past decades have seen important progress in constraining the physics of GC formation from a variety of directions. In this article, we discuss the latest constraints obtained from studies of present-day GC populations, the formation of young massive clusters (YMCs) in the local Universe, and the observed, large-scale conditions for star and cluster formation in high-redshift galaxies. The main conclusion is that the formation of massive, GC progenitor clusters is restricted to high-pressure environments similar to those observed at high redshift and at the sites of YMC formation in the local Universe. However, the correspondingly high gas densities also lead to efficient cluster disruption by impulsive tidal shocks, which limits the survival of GCs progenitor clusters. As a result, the long-term survival of GC progenitor clusters requires them to migrate into the host galaxy halo on a short time-scale. It is proposed that the necessary cluster migration is facilitated by the frequent galaxy mergers occurring at high redshift. We use the available observational and theoretical constraints to condense the current state of the field into a coherent picture of GC formation, in which regular star and cluster formation in high-redshift galaxies naturally leads to the GC populations observed today.