Yohan Brunebarbe of the University of Bordeaux presents this research talk at the Institute for Advanced Study, part of a workshop on Hodge theory and o-minimality. He takes up Shafarevich's question of whether the universal cover of a smooth projective variety is always holomorphically convex, meaning it admits a proper holomorphic map to a Stein space. He reviews the linear case, settled by Eyssidieux, Katzarkov, Pantev, and Ramachandran using non-abelian Hodge theory, then presents a generalization to non-compact algebraic varieties developed jointly with Ben Bakker and Jacob Tsimerman. The talk assumes graduate-level familiarity with complex algebraic geometry and fundamental groups, and proceeds through the technical machinery used to construct holomorphic functions on these universal coverings. Delivered at Simonyi Hall, it is a specialist seminar talk aimed at researchers working in Hodge theory, complex geometry, and related fields rather than a general audience.