We are trying to provide a collection of previous successful grant applications and reviews. The idea is that future students can have a look at previous applications in one consolidated place to have some idea of what is being looked for. Old applications float around, and there are some websites out there that have some old applications posted, but having applications specific to our fields might be useful.
In your first two years as a graduate student you can apply for the National Science Foundation's Graduate Research Fellowship.
If students are between the ages of 8 to 25 they can apply for the National Geographic's Young Explorers Grants (YEG) which offer opportunities to pursue research, conservation, and exploration-related projects consistent with National Geographic's existing grant programs, including: the Committee for Research and Exploration (CRE), the Expeditions Council (EC), and the Conservation Trust (CT).
In later years students can apply for the Doctoral Dissertation Improvement Grants. These grants provide partial support of doctoral dissertation research to improve the overall quality of research. Allowed are costs for doctoral candidates to participate in scientific meetings, to conduct research in specialized facilities or field settings, and to expand an existing body of dissertation research.
Other great opportunities for funding are the Leakey Foundation's general research grants and the Wenner Gren Foundation's Dissertation Fieldwork Grants.
Throughout their career individuals can apply for the Fulbright fellowships for funding research in certain foreign countries.
Stony Brook maintains a database with funding opportunities for researchers, which is a useful place to look for opportunities specific to your field of research.