Is it just nationally, or internationally? Large establishments only, or even small, family-run parks? Do they get a free tour too, and free encounters? (OK, that latter question was just me daydreaming a bit about how lucky they must be.

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I have friends who are zoo keepers and whenever we have travelled together, they have notified their department manger or curator and have had these people contact the zoos we were planing to visit in advance and provided us with a point of contact at the zoo for us to connect with.
On arrival we have gone through the general admission area and then had our contact notified by radio of our arrival.
You are going to love this nanoboy, what happens next is that we are given a behind the scenes tour of the zoo including animal encounters.
These encounters are very random and dependant on many many things including my friends speciality [usually Gorillas]area department staffing levels, wether or not an animal is new or recently placed into an enclosure, if an area was being cleaned ect ect ect.
I actually had no idea how sensitive many animals were to having behind the scenes visitors, it seems some animals could not care less and others it was out of the question.
It seems that the communication between zoos on husbandry and experience is actually not as free flowing as you might expect. The conversations l have listened to [so rarely been actively involved in] have always surprised me. l would have thought that communication between zoos, and or specific staff managing specific species were talking all the time i get the idea that they are not!
I was once listening to a very interesting conversation about the extra benefits of keepers travelling with animals in relocations, that yes most of the animals appreciated the consistency of contact, however the sharing of experiences between keepers was actually considered to equally or more important.
To be clear on just how much of a zoo geek l am, l can honestly say that the days spent going behind the scenes of a zoo with a zoo keeper friend and the assorted staff of a visited zoo [ btw you get passed around different keepers rarely staying with any one person ] have been the best days of my life.
Some highlights,
Dinner with the owners and a private tour of Gorilla Haven
Patting a baby Giraffe at Taronga Zoo.
Bottle feeding a baby gorilla at Columbus zoo.
Feeding penguins at Sea World Orlando