Universality in numerical computations with random data. The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time -- i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components -- the sample average and the sample variance -- the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms as well as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own. par See also the related work of the first two authors with {it C. W. Pfrang} [Math. Sci. Res. Inst. Publ. 65, 411--442 (2014; Zbl 1326.65050)].