Note: This is a revised version of 0251. This version corrects some errors in the section on topological symmetric spectra.
The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum.