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.