One more time: autoformers?

Not sure if this OP belongs in cables, speakers, or amps. Since I listed cables first -- it's cables. There are already some old threads which touch on the use of autoformers. Seems like most of the threads come up in the context of OTL and SET amps.

My Qs are: What are benefits and downsides to using autoformers? How does one know if an autoformer would be a good match for his/her rig?

As with most things in life, I suspect that autoformers must have downsides. Otherwise, everyone would use them. Open and intellectually honest responses are welcome.

Thanks?