This talk will concern two topics.  The first topic is the applications of Riemann--Hilbert (RH) problems.   RH problems provide a powerful and rigorous tool to study many problems in pure and applied mathematics.  Important problems in integrable systems and random matrix theory have been solved with the aid of RH problems. RH problems can also be approached numerically with applications to the numerical solution of PDEs and the sampling of random matrix ensembles.  The resulting methods are seen to have accuracy and complexity advantages over previously existing methods.  The second topic is recent progress on the statistical analysis of numerical algorithms.  In particular, with appropriate randomness, the fluctuations of the iteration count of numerous numerical algorithms have been demonstrated to be universal, see Pfrang, Deift and Menon (2014).  I will discuss simple algorithms where universality is provable and the wide persistence of this phenomenon.