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.
Wikipedia, Riemann-Hilbert correspondence
Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les -modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026
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.