Various versions of the Riemann-Hilbert problem for linear differential equations

A counterexample to Hilbert's 21st problem was found by Boli-brukh in 1988 (kind published in 1989). In the further study of this problem lie substantially developed the approach using holomorphic vector bundles and meromorphic connections. Here the best-known results of the past that were obtained by using this approach (both for Hilbert's 21st problem and for certain generalizations) are presented.