Finding Digitized Primary Sources

Great Britain and Ireland:

Times of London and Sunday Times
The Guardian and The Observer, 1791-2003
17th-18th Century Burney Collection Newspapers
Eighteenth Century Journals: Newspapers and Periodicals, 1685-1835
The Illustrated London News Historical Archives
The Irish Times
Irish Historical Newspapers
Times Literary Supplement Historical Archive (in Gale NewsVault)
Picture Post Historical Archive, 1938-1957
The Punch Historical Archive, 1841-1992
19th Century U.K. Periodicals
British Library 19th-Century Newspapers
Daily Mail Historical Archive, 1896-2004
Economist Historical Archive
Financial Times Historical Archive
Independent Digital Archive, 1986-2016
Gale NewsVault
Service Newspapers of World War II

Continental European Newspapers:

Newspapers.com World Collection
Europeana Newspaper Collection -- digitized historic newspapers from the collections of many of Europe's National Libraries
Digital French-Language Newspaper Portal
Mercure de France, 1672-1810
RetroNews -- 400 titles of the French-language press published from 1631 to 1950  
ANNO (Austrian Newspapers Online)
Communist Historical Newspaper Collection
Service Newspapers of World War II

Latin American Newspapers

Latin American Newsstream

World Scholar: Latin America and the Caribbean

Latin American Anarchist and Labour Periodicals Online

Newspapers.com World Collection

Caribbean Newspapers, 1718-1876

Independent and Revolutionary Mexican Newspapers

Africa:

African Newspapers
SA Media (1977 -- ), South African newspapers
AllAfrica.com (1997 -- )
Africa Confidential (1996 -- )

Asia:

South Asian Newspapers
Chinese Newspapers Collection, 1832-1953 (ProQuest Historical)
Chinese Newspapers: Tabloids, 1897-1949
The Times of India
Newspapers.com World Collection
Historical Japanese Newspapers
Trans-Pacific Weekly Review (Japan)

Asia and Africa:
Service Newspapers of World War II