Have you ever had a bird you can take out of the cage without fly away?

I was again watching some cool videos of pet birds. Their owners do not even use cages for them, because these birds will not get away. They are so attached to their owners that the owners can take them out for a walk on the shoulder or on the arm. I am totally jealous.

When I was a kid, I have had some parakeets. The best I could do was to take one of them out and put him on my head, although I had to close all doors and windows before I could feel safely to do that.

In one of the bird videos I have seen, even parakeets could be attached to the owners. I wonder what the trick is. I'd kill for a bird who can be with me everywhere I go without having to use a cage.

Have you ever had a bird you can take out of the cage without fly away?
What species is it?