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Ars Technica reports on a May 24 phys.org press release/precis on a University of Glasgow paper in Journal of Spacecraft of Rockets about a solid rocket intended to consume itself during use rather than require staging to improve its mass fraction (hence "autophage").

The paper, which I apparently have access to at work but not from home, mentioned that there were previous efforts toward the same goal, but it's not currently clear to me how those went. This one currently seems to require a separate nozzle and has some tendency to explode.

What other autophage rockets have been built, and what stage of development did those reach?

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    $\begingroup$ I imagine that autophage rockets aren't attractive to develop, because solid rocket applications tend to emphasize cost efficiency more than mass efficiency, and developing and producing an autophage has to be more expensive than a traditional solid stage. $\endgroup$ – Russell Borogove Jun 2 '18 at 1:45
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    $\begingroup$ Very cool!! Would you consider adding some of this as an answer to Burn 1st stage structural material as fuel? $\endgroup$ – uhoh Jun 2 '18 at 2:56
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The patents cited by that paper (1, 2, 3, 4) show no evidence of construction and testing. But a laboratory-scale model was tested in 2015, says this paper (paywalled except for the first page).

The J.S.R. paper's set of test firings was even covered by The Economist. The propellant was half a meter long and burned for a few minutes, with a chamber pressure of about 500 kPa, just a few per cent of the chamber pressure of SpaceX's Raptor engine. Because these tests ended "due to fuel starvation or, sometimes, another failure mechanism," one might say that this state of the art is still at the Jebediah Kerman stage.

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    $\begingroup$ There's a wealth of information in those references! Some more of it should probably end up in the answer here, but thanks for the attention to a year-old question $\endgroup$ – Erin Anne May 16 at 1:01

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