Can ReSharper run unit tests automatically

This has been available since dotCover 10. See the dotCover documentation for details.

This adds a new panel "Continuous Testing Session" as well as a new status icon in the gutter.

Note that Visual Studio also has this feature, known as Live Unit Testing.

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Not possible with Resharper at the moment, you will need something like NCrunch that runs your unit tests continuously in the background, highlighting code that breaks them as you write it and fails your tests.

Edit: At the time of my response it wasn't possible to do this with ReSharper but now in Version 10 it is, see Drew Noakes's answer. You could still give NCrunch a try as it continuously runs your tests in the background even without doing an explicit save.