This article quotes Teresa Iglesias, a former Grass Fellow at MBL who worked with Senior Scientist Roger Hanlon to identify a cyclic, REM sleep-like state  in cuttlefish (published here).

Young jumping spiders dangle by a thread through the night, in a box, in a lab. Every so often, their legs curl and their spinnerets twitch — and the retinas of their eyes, visible through their translucent exoskeletons, shift back and forth.

“What these spiders are doing seems to be resembling — very closely — REM sleep,” says Daniela Rößler, a behavioral ecologist at the University of Konstanz in Germany. During REM (which stands for rapid eye movement), a sleeping animal’s eyes dart about unpredictably, among other features.

In people, REM is when most dreaming happens, particularly the most vivid dreams. Which leads to an intriguing question. If spiders have REM sleep, might dreams also unfold in their poppy-seed-size brains? Read rest of the story here.

Source: Do spiders dream? about cuttlefish? Bearded dragons? | Knowable Magazine