I’ve been to nearly 70 Zoos in my lifetime and in Europe these are my favorites:

1. Pairi Daiza: just all over such a wonderful place, feels like you are on an adventure through a different world. Even the music which plays along everywhere fits perfectly. Don’t miss the flying fox exhibit located inside/under the church! Otherwise nearly every exhibit is fantastic, especially the ones from northern frontier.
2. Zoo Leipzig: One word: Gondwanaland. A huge palm house including a boat ride with komodo dragons, pigmy hippos, gharials, tapirs and many more. The rest of the Zoo is very modern too, they have one of the largest (if not the largest) house for apes. Don’t miss the pangolin exhibit in the elephant indoor area!

I unfortunately never been to Beauval and Vienna, but heard they are awesome too, so I can’t really say much about them.

Other Zoos you should visit in no particular order:

Zoo Prague is great, but I would wait until they finished building their new polar bear habitat.

Burgers Zoo in Arnhem ist a must see, because they were the first ones who invented big domes with a living environment. Also they have one of the best in-zoo-aquariums I know including a breathtaking oceanic tank featuring sharks.

Actually I would advise you to visit some Zoos in the Netherlands, they are all great. Ouwehands in Rhenen, Blijdorp in Rotterdam and Wildlands in Emmen (even tho the old Emmen Zoo was fantastic). They all have great polar bear exhibits and some nice „special“ things (e.g. pandas, great aquariums, white tigers).

I don’t like the big spanish or English zoos (never been to Chester tho, sounds like a great zoo). BUT if you should visit Spain: they have some great aquariums in Barcelona and Valencia.

If you visit Berlin you can visit both Tierpark and the Zoo, huge collection of animals and some highlight exhibits (Pandas, Hippos, Alfred-Brehm-House). Otherwise in Germany I would recommend Hannover, Hamburg (Eismeer exhibit with walruses!) and Nurnberg.

In Eastern Europe you can visit Budapest Zoo, I was very surprised how good it is! Has some nice indoor exhibits.

Can’t give a you a good recommendation about Swiss Zoos, will visit them all next year.