The most basic form of the Riemann–Hilbert correspondence states that the category of flat vector bundles on a suitable space is equivalent to the category of local systems.
While Hilbert's 21st problem has a negative solution, there is a generalized sheaf-theoretical formulation which leads to an equivalence of categories discovered by Mebkhout and a bit later also by Kashiwara.
Alain Connes, Matilde Marcolli Noncommutative Geometry, Quantum Fields and Motives
Wikipedia, Riemann-Hilbert correspondence
Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026
Andrea D'Agnolo?, Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic D-modules, arxiv/1311.2374
> The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin’s work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.
Generalization to flat infinity-connections via the dg-nerve and iterated integrals is discussed in