UK and Ireland Newspapers

• Richard Heaton provides a very useful Index of British and Irish Online Digitalised Newspapers - lists of all the titles of British and Irish Newspapers (31 different collections covering 17th to 21st Century) now available online, providing links, subscription details and dates covered.
• Wikipedia provides a very extensive List of Online Newspaper Archives. (You'll need to scroll down to find the United Kingdom section.)
• The British Library provides, in collaboration with FindMyPast, online (subscription) access, to their newspaper collection, through the British Newspaper Archive service. (As of July 2016 this had 656 titles online - see their coverage and video guide.)
•  Web News - the British Library's archive of "selected UK news sites on a daily or a weekly basis since September 2013".
• The British Library's News Media Collection Guides.
• Newsplan - was a co-operative programme (involving between public libraries, the British Library, the National Library of Scotland, the National Library of Wales, the National Library of Ireland, and the newspaper industry) for the microfilming and preservation of local newspapers and for making them accessible to users.
• The London, Edinburgh, and Belfast Gazettes are the official newspapers of record in the United Kingdom. As well as the current editions, the website includes searchable archives of the complete gazettes from WW1 and WW2 containing medal awards, army and navy commissions, promotions and much more.
• The Newspaper Society's online Regional Press Database lists regional and local newspapers in the UK.
• British Media Online - All the national and regional newspapers published online.
• UK Press Archive - includes Daily Mirror, Daily Express, The Church Times, and more (accessible to subscribing libraries and UK further and higher education institutions.)
• Bev Edmonds has transcribed these Extracts from Various Late 19th & Early 20th Century Newspapers (archived copy).
• Ancestry's Newspapers & Periodicals Collection - includes The Times, The Gentleman's Magazine Library, and many more (subscription)
• UK and Ireland, Obituary Index, 2004-Current - a free search facility provided by Ancestry of obituaries collected automatically from recent online editions of newspapers.
• Historical Newspapers - a (subscription) service incorporating The Official Index to The Times, Palmer's Index to the Times, The Historical Index to The New York Times, and Palmer's Full Text Online, 1785-1870.
• The Nineteenth-Century Serials Edition (ncse) is a free, online edition of six nineteenth-century periodicals and newspapers: The Monthly Repository and Unitarian Chronicle, The Northern Star, The Leader, The English Woman's Journal, The Tomahawk, and The Publishers' Circular.
• Richard Heaton's Newspaper Collection - "Extracts and full Transcripts of mainly English and Irish, Georgian and early Victorian Regional Newspapers"
• Internet Library of Early Journals - a project that digitised 20 years or more of The Annual Register, Blackwood's Edinburgh Magazine, The Builder, The Gentlemen's Magazine, Notes and Queries, Philosophical Transactions of the Royal Society (free). 
• Gale Historical Newspapers - provides simultaneous searching  across a number of British Library resources: The 17th and 18th Century Burney Collection, The 17th and 18th Century Nicols Newspapers Collection, 19th Century UK Periodicals, British Library Newspapers 1800-1900, and many more.
• The Waterloo Directory of English Newspapers and Periodicals (1800-1900) - "a subject-inclusive, language-inclusive bibliography of 73,000 publications, 68,000 personal names".
• Australia Trove - a large searchable collection of scanned and digitised historical Australian newspapers.
• The Guardian and Observer digital archive - "1.2 million replica pages of the Guardian and Observer dating back to 1791" (subscription).
• Newspapers in Family History, by Richard Ratcliffe.