Rees algebras of ideals of maximal minors of generic matrices are very well understood. In this talk we deal with the case of sparse matrices with two rows and with the case of two by two minors of generic matrices. We investigate the defining ideals of these algebras, and in the first case we prove that they are Koszul and have rational singularities or are F-rational, respectively. This is a report on joint work with Ela Celikbas, Emilie Dufresne, Louiza Fouli, Elisa Gorla, Kuei-Nuan Lin, and Irena Swanson and with Hang Huang, Michael Perlman, Claudiu Raicu, and Alessio Sammartano.

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Rees Algebras Of Ideals Generated By 2x2 Minors