Some individual awards exist, but the majority of government grants are awarded to eligible nonprofit organizations. The following are some sites and publications that are helpful in seeking government funding.
Catalog of Federal Domestic Assistance
Grants.gov - searchable databases of >1,000 discretionary grant programs of 26 federal agencies
More Specific Grants
Government Grants and Loans - Grants, Loans, and Other Assistance; Management and Operations;and Tax Information
MOLIS - Minority On-Line Information Service
National Endowment for the Arts - apply, manage or see a list of recipients
National Endowment for the Humanities- apply, manage, get news or see the projects
National Science Foundation - apply, manage