Here fishy fishy!...... fishy?

A certain question has haunted me for some time. In all the years I've spent fishing with my dad, granddad, father-in-law and others, I've never been sure, and never bothered to ask.
I probably figured I would look ridiculous, but since when has THAT stopped me from doing something. (I can hear Meg and mom right now turning to each other and high-fiving, along with an "amen on that one!")
So, do fish actually sleep?
I know, not very profound. But I HAD TO KNOW this morning! So after 3.2 seconds typing the question into my google search box, and another 3.2 clicking over to the linked page, I had my answer.
So, yes and no.
Yes, they sleep.
No, they don't lean over on their fins and start snoring, ignoring th world around them. That would be me. Without fins of course.