An organization I was at in the past attempted to use them as a replacement for sessions, which turned out to be a terrible idea as I suspected it would.
I've found that arbitrarily re-inventing the wheel because a new thing becomes popular should be done deliberately and with great caution. More generally - I think it's important to look for solutions to fit a specific problem, not problems to fit a specific solution.
However, back to JWTs - I'm currently using them for authorization in an EXTREMELY high traffic websocket server implementation. It's really nice because it's short duration (the ones I am issuing have a expiration of 60 seconds), and allows the service to operate entirely within memory except for interacting with a Kafka cluster.