The theory of testing statistical hypotheses aims not only at proposing suitable tests, but also (and mainly) at deriving their optimal procedures under certain conditions. Any type of data contamination however requires non-standard approaches to testing. This talk, which is intended for non-statisticians, will very briefly overview principles of several recent results, on which I participated. To be specific, particular results include: Minimax tests under measurement errors, Locally optimal tests based on sequential ranks, Tests based on ranks of interpoint distances among high-dimensional observations, Symmetry tests based on robust multivariate estimators.