oEmbed
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oEmbed is a JSON/XML-based format for providing information to embed a page or summary thereof, like a link-preview.
- See HTML-based alternative: rel-embed
History
Was created in 2008 at the general posting silo Pownce to enhance the link sharing on that platform.
Criticism
Requires Extra Endpoint
oEmbed requires that any site support it support another URL (the oEmbed endpoint) to provide information that is otherwise available in the page itself.
This appears to be a design decision to reduce the parsing effort for the including sites, some implementations do not parse the embedded page at all and rely on a centralized list of endpoints.
embedded site provides markup
The site being embedded provides the markup to construct the preview. While this gives the site being embedded a lot of control over the display, this might clash with the wishes of the embedding site, accidentially conflict with existing CSS and pose security issues. This in turn makes whitelisting of sites attractive, which means non-major sites are less likely to be included.
direct embeds are a privacy/tracking issue
Since content often is included directly from the source site, this leaks information about your visitors to this site and can be a problem for your privacy/Do Not Track policy.
Support
As of late 2017, oEmbed is supported by many major silos (see central list of supporting sites). WordPress provides an oEmbed endpoint by default as well, many other CMSes at least can act as consumers and embed content from oEmbed-enabled sites.
Discussion
Some discussion of oEmbed vs. a simpler link-preview approach has occurred in IRC:
oEmbed to enable reposting
Reposting can bring along copyright issues. It also means there may be copies of your content floating around that you can no longer edit or remove. 
Martijn van der Ven is thinking about adding an oEmbed endpoint to his blog to enable reposting in a way that he should stay in control over the copies: the oEmbed endpoint output can be updated at any time.