Let ξ be a real-valued random variable of mean zero and variance 1. Let M n (ξ) denote the n × n random matrix whose entries are iid copies of ξ and σ n (M n (ξ)) denote the least singular value of M n (ξ). The quantity σ n (M n (ξ))2 is thus the least eigenvalue of the Wishart matrix $${M_nM_n^\ast}$$ . We show that (under a finite moment assumption) the probability distribution n σ n (M n (ξ))2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of M n (ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices. Our approach is motivated by the general idea of “property testing” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics.