The inverse scattering transform (IST) is used to solve the Cauchy problem for integrable nonlinear partial differential equations on the line.  Matrix Riemann-Hilbert problems (RHPs) are a key component in the IST.  Historically, RHPs have made the IST amenable to rigorous asymptotic analysis with the Deift-Zhou method of nonlinear steepest descent.  More recently, techniques for oscillatory singular integral equations have been employed to solve RHPs numerically and compute the IST.  Importantly, nonlinear dispersive evolution equations can be solved numerically without any need for time-stepping.  Errors are seen to be uniformly small for arbitrarily large times.  Combining this approach with the so-called dressing method allows for the computation of a wide class of non-decaying solutions.