In the 60s/70s, an EE/CS guy named Leon Chua did some clever math that suggested we could build transistor-like devices that had a non-linear relationship between charge and flux and called it a memristor. He didn't build one, and as far as I know it's not something he's spent all that much time on since.
In the 90s somebody at HP must have mentioned memristors in hearing range of their marketing department and suddenly HP was going to have a product to market in a few years.
Yeah. Never happened. Nobody's done it.
In theory, such a device could allow us to do things like build extremely low power non-volatile memory (a flash replacement, basically). A lot of marketing hype has talked about completely replacing computer RAM with non-volatile memristor-backed memory, but again, neither it nor anything like it has happened. But as it turns out, battery-backed DRAM covers a multitude of sins anyway.
The actual debate over physical possibility is because it's too low-power, to the point that an actual implementation might violate lower theoretical limits on the amount of energy required to change the state of a bit. In other words, it might violate the second law of thermodynamics, which is something of a problem.