There are a range of funding opportunities for contract research and development. See the Sponsor page for a discussion of some of the sponsors of CRAD funding and the Grants page for a discussion of where to look for grants.
The primary difference between CRAD and grant opportunities is that CRAD opportunities allow a company to charge a fee. There are no fees allowed on grants, though they can still be an excellent way to get funding.
Government funding for CRAD opportunities are normally published under Broad Agency Announcements (BAAs). BAAs over $25,000 by law have to be published on the Federal Business Opportunities web page, www.fbo.gov. Government grants for R&D are published through grant notices on agency websites and on the Grants.Gov web page.
Other Agencies, such as NASA, publish calls for research under different names. For NASA it is the NASA Research Announcements (NRA). Grants are available from a range of agencies, such as the National Science Foundation, National Institute of Health.
Small business have additional funding opportunities, such as through the Small Business Innovative Research and Small Business Technology Transfer and the Rapid Innovation Fund, as well as small business set asides. Requests for Information can also lead to funding.