Wikipedia has a list of free and subscription digital online newspaper archives. Most are scanned from microfilm into pdf, gif or similar graphic formats and many of the graphic archives have been indexed into searchable text databases utilizing optical character recognition (OCR) technology. Some newspapers do not allow access to the OCR-converted text until it is proofread. Older newspapers are still in image format, and newer newspapers are available as full text that can be cut and pasted. Most text is in ASCII, some are using Unicode for diacritical marks not available in ASCII. Google now indexes many newspaper archives.
Historical Newspapers Online is a collection of newspapers listed by states and is part of the Penn Libraries collections.
Chonicling America is a collection of over 600 newspapers from 30 states published between the 1830s and 1922, and it continues to grow rapidly. Backed by the Library of Congress and the National Endowment for the Humanities, these newspapers are all freely available. Although each state on the newspapers by state listing indicates if Chronicling America includes newspapers from it, this site also allows you to search all Chronicling America newspapers simultaneously.