Reference request: Riemann-Hilbert problems I am trying to learn more about the applications of complex analysis in solving spectral problems and came across applications of theory built around Riemann-Hilbert problems. So far I have only read about these problems in Novikov's Theory of Solitons, but would like a text that covers the underlying analysis better instead of treating applications without rigorous proofs. 
Can anyone provide me with a decent text? Thanks! 
 A: It depends on you backround in complexe analytic geometry and algebraic geometry, but anyway, as modern analysis uses more and more sheaves techniques etc...
First there a nice "classical motivation of RH" by Brian Conrad :
http://math.stanford.edu/~conrad/papers/rhtalk.pdf
Second you would want to see proofs of the RH correspondance for regular holonomic $\mathscr{D}$-modules. Whatever it may be said, the first proof who was given was given by Zogman-Mebkhout and you can find it in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory that you can easily find yourself on the internet.
For Kashiwara's proof you can find it here :
http://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=20&iss=2&rank=4
For the irregular holonomic case, you can directly look at a Kashiwara's talk here :
https://www.youtube.com/watch?v=JWVvC8yEvP4
(you will have difficulties to understand sometimes, as Kashiwara is not the japanese speaking english the best)
You can find the corresponding paper here (quite recent, I don't actually have the most recent version with me) :
http://arxiv.org/pdf/1311.2374.pdf
For accounts on $\mathscr{D}$-modules and related notions involved in previous references, you can find a proper exposition (in french) in Elements de la théorie des systèmes différentiels géomtriques, Séminaires et Congrès (SMF) number 8. You can find the whole book here :
http://www.emis.de/journals/SC/2004/8/html/
A: It has been a long time since you asked your question, but I guess it may help other too ("mejor tarde que nunca" like we say in Spanish).
One of the most important applications of the (analytic) Riemann-Hilbert problems is in the theory of Orthogonal Polynomials. There's a fantastic little book by
1) Deift - Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach,
which explains how to use RHPs to extract large-n asymptotics of orthogonal polynomials.
Furthermore, there's also a book 
2) Edited by Harnad - Random Matrices, Random Processes and Integrable Systems,
in which there are two sections by Pavel Bleher and  Alexander Its on the Riemann-Hilbert problem and its uses in mathematical physics, complex analysis and asymptotics.
For more reference see the papers these notes point at.
Remark: I want to point that in the literature exists two things called Riemann-Hilbert problems, one of them being what Olórin mentioned in their response - the 'algebraic' Riemann-Hilbert problem-, and another being the one stated here -the analytic Riemann-Hilbert problem-.The two are of course related, since they have to do with the inverse monodromy problem of linear differential equations, however, the two worlds are currently very distinct in approach and goals. 
