The Hubble Space Telescope (HST) has been instrumental in the discovery of large numbers of extragalactic young massive star clusters (YMCs), often assumed to be proto-globular clusters (GCs). As a consequence, the field of YMC formation and evolution is thriving, generating major breakthroughs as well as controversies on annual (or shorter) time-scales. Here, I review the long-term survival chances of YMCs, hallmarks of intense starburst episodes often associated with violent galaxy interactions. In the absence of significant external perturbations, the key factor determining a cluster’s long-term survival chances is the shape of its stellar initial mass function (IMF). It is, however, not straightforward to assess the IMF shape in unresolved extragalactic YMCs. I also discuss the latest progress in worldwide efforts to better understand the evolution of entire cluster populations, predominantly based on HST observations, and conclude that there is an increasing body of evidence that GC formation appears to be continuing until today; their long-term evolution crucially depends on their environmental conditions, however.