This book is drawn from the recent literature on the asymptotic behavior of random permanents and random matchings. In particular, the authors present an elegant connection between the problem of an asymptotic behavior for a certain family of functionals on random matrices and the asymptotic results in the classical theory of the so-called U-statistics -- objects of fundamental importance in the non-parametric statistical inference. This book is self-contained and accessible to any mathematics, statistics or engineering graduate student who has taken basic introductory courses in probability theory and mathematical statistics. Dr.Grzegorz A. Rempala is a Professor of Statistics in the Department of Mathematics at the University of Louisville in Louisville, KY. Dr. Jacek Wesolowski is a Professor of Mathematics and Associate Dean for Research at the Faculty of Mathematics and Information Science, Warsaw University of Technology in Warsaw, Poland. The volume is a result of the authors collaborative effort initiated at the IMA during the Institute's 2003/04 annual program on 'Probability and Statistics in Complex Systems: Genomics, Networks, and Finance Engineering'.