On the Singularities of Complete Holomorphic Vector Fields in Dimension Two

For a germ of singular holomorphic vector field on a complex manifold to be the local model of a complete one, it is necessary for its solutions to be univalent (and not multivalued). Rebelo formalized this local obstruction to completeness through the notion of semicompleteness, which he started studying some twenty-five years ago; it has been investigated ever since by various authors. We here review this notion, from the foundational definitions to some local and global results, with a special emphasis on manifolds and analytic spaces of dimension two.