Globular clusters (GCs) are fascinating objects nearly as old as the Universe that provide insight on a large variety of astrophysical and cosmological processes. However, their formation and their early and long-term evolution are far from being understood. In particular, the classical paradigm describing GCs as large systems of coeval stars formed out of chemically homogeneous material has been definitively swept away by recent high-precision spectroscopic and deep photometric observations. These data have provided undisputed evidence that GCs host multiple stellar populations, with very peculiar chemical properties. In this talk, I will review the properties of these multiple populations, before presenting the different scenarios that have been proposed to describe their formation. I will focus on the (many) current theoretical issues and open questions.