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The present paper gives an overview of the recent developments in thedescription of critical behavior for Hamiltonian perturbations of hyperbolicand elliptic systems of partial differential equations. It was conjectured thatthis behavior can be described in terms of distinguished Painlev\'etranscendents, which are universal in the sense that they are, to some extent,independent of the equation and the initial data. We will consider severalexamples of well-known integrable equations that are expected to show this typeof Painlev\'e behavior near critical points. The Riemann-Hilbert method is auseful tool to obtain rigorous results for such equations. We will explain themain lines of this method and we will discuss the universality conjecture froma Riemann-Hilbert point of view.

The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems