Members of the library can use the online PressReader service, which gives free access to thousands of worldwide newspapers and magazines, including UK titles. Download up to 20 titles a day via the app or unlimited streaming. 100 UK and US popular magazine titles are also available from RBdigital.
The National Library of Scotland provides some newspaper resources online to members. Within the Reading Room itself you can access the British Newspaper Archive with millions of pages from British newspapers from the 1700s to the 1900s are available online . Outwith the reading rooms, you have to buy credits to view, print and download articles and pages.
The National Library also has a large collection of printed newspapers, but it is best to regard this as something to try when you have exhausted all other convenient resources. The collections include a representative sample of the UK regional press and a selection of overseas newspapers. Some newspapers are held on microfilm to help reduce the handling of rare and fragile material.
Most of Edinburgh University's newspaper resources are provided online, and will only be available to Edinburgh Unviersity staff and students. However, if you are struggling in the search for a historical resource, as a last resource it might be worth contacting them, as they may have print or microfilm copies.
If you are doing research which absolutely demands access to scarce historical newspaper material, you just might have to consider a visit to London or Boston Spa to the British Library. If you do, please make sure you have read up on their service before you go, or you may have a wasted trip:
Information at May 2018: please check their website for the current picture
The Newsroom, our new dedicated news Reading Room at the British Library in London, is the focal point for researching the news collections. The Newsroom provides access to newspapers in print, microfilm and digital formats. The British Library operates a “surrogate first” policy for its newspaper collection. A surrogate copy will be a microfilm or digital facsimile of the original newspaper and the surrogate will be the Reading Room access copy. Where a surrogate copy exists, the original print newspaper will be labelled as “restricted” on Explore the British Library. This policy is designed to preserve the fragile original newspapers in our care.
Readers may order non-surrogated print newspapers to both the Newsroom at St Pancras and to the Boston Spa Reading Room. Print newspapers stored in the National Newspapers Building take 48 hours to arrive at the Newsroom or the Boston Spa Reading Room. Microfilm newspapers are delivered to the Newsroom at St Pancras within 70 minutes of request. Microfilm is currently not available to order to the Boston Spa Reading Room.
There is lots of useful information on their News Media page.
They also provide a helpful guide to Researching Newspapers.