The Shafarevich conjecture asserts that the universal cover of a smooth projective variety \(X\) is holomorphically convex. This conjecture holds true for curves (by the Poincaré uniformization theorem). The paper under review is a survey on the recent developments in understanding the Shafarevich conjecture. The author discusses first the results of Kollár and Campana concerning the birational existence of the Shafarevich maps. Then one presents certain approximations of the Shafarevich conjecture for special groups, e.g. a theorem of Jost-Yau and Siu which says that if the fundamental group of a curve of genus \(g\) is a quotient of \(\pi_1(X)\), then \(X\) has a morphism onto a curve of genus at least \(g\). Next one considers the nonabelian Hodge theory (Corlette, Hitchin, Simpson) and its \(p\)-adic analogue (Gromov, Schoen). As an application, one proves that the Shafarevich conjecture holds true for surfaces with linear fundamental group. In the last part the author proposes an example which reduces the Shafarevich conjecture to some difficult unsolved group-theoretic questions.
For the entire collection see [Zbl 0882.00033].