Getting access to scientific papers behind a paywall
You're reading a news article about a new medical procedure that might make pancreatic cancer significantly more survivable. The news article seems a bit vague and hyperbolic, so you'd like to read the study they're naming. When you look it up, a paywall appears, asking you pay £28 to read the article or £508 for a yearly subscription to that scientific journal. This is not helpful to the interested amateur.
What are the alternatives?
Open Access content
These services are fully ethical and free ways of finding full PDF or HTML content for paywalled papers:
Google Scholar - Simple search engine for scholarly literature.
Unpaywall - Browser extension linking to free versions of paywalled papers.
Open Access Button - A search engine as well as a browser extension to find papers.
The grey method
N.B. This service values open access to the world's scientific findings higher than respecting the intellectual property of publishing companies.
Sci-Hub - Search engine with a very comprehensive collection of articles..
When Sci-Hub cannot be found via this link, you can find it at alternative urls listed at https://sci-hub.41610.org/.
Crowd-sourced
On Twitter, if you use the search bar to query a combination of an article title and the hashtag #icanhazpdf, you may find the article has been previously requested and a response with the article PDF or HTML source. If not, you can submit a request yourself. Just follow the examples you can view when you search for #icanhazpdf.
When all else fails
When you contact the author(s) of a scientific paper directly (e.g. via email) and kindly request a PDF copy of it, they may very well be happy to share it with you.