Tubeless - Tell me about the Pros, Cons and options

A long long time ago, I puchased tubeless tyres. They were called Green Tyres and they had no air in them and they made the rims of my Raleigh Mustang go a funny shape.

After that I never bothered with tubeless again.

Since my passion for cycling has been rekindled and my OCD has kicked in again, I'm trying to get my head around tubeless tyres and the options available to me.

Anybody want a go at explaining? Reading promotional websites etc only goes so far in understanding the options