arXiv.org, hosted by Cornell University, was among the first open access repositories. It currently hosts hundreds of thousands of scholarly papers and datasets in Physics, Mathematics, Computer Science, Quantitative Biology, and more.
Subject-specific repositories exist for most subject areas, find examples here: http://libguides.asu.edu/openaccessresources/home
Some scholars opt to post their work to individual repositories, either because their institution does not provide one or their subject does not yet have a comprehensive one, or because they hope to maintain more control.
Academia.edu and ResearchGate are two prime examples.
Many universities provide a repository in which students, faculty, and researchers can post and link to their publications, datasets, and other scholarly work. Some of these repositories also include electronic thesis or dissertations and some may include special collections materials. An updated list of academic repositories is here: http://www.opendoar.org/countrylist.php