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We describe two recently proposed randomized algorithms for the construction of low-rank approximations to matrices, and demonstrate their application (inter alia) to the evaluation of the singular value decompositions of numerically low-rank matrices. Being probabilistic, the schemes described here have a finite probability of failure; in most cases, this probability is rather negligible (10⁻¹⁷ is a typical value). In many situations, the new procedures are considerably more efficient and reliable than the classical (deterministic) ones; they also parallelize naturally. We present several numerical examples to illustrate the performance of the schemes.