Mainland Europe is quite big, so I will focus on zoos that lie in the same corner as Antwerp
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As
@oflory mentions, Pairi Daiza will receive golden snub-nosed monkeys at some point next year, so is interesting. Their enclosure quality is often not the best, but it is an intriguing zoo that is quite unique
A must see for you seems to be Apenheul, which houses >30 primate species in good to very good (walk-through) enclosures. By car it is some 2-3 hours from Antwerp. It would also be a shame to leave out Burgers' Zoo if you are in that area. It might not have a huge primate collection (but it has Peruvian squirrel monkeys, pig-tailed macaques & golden-bellied mangabeys), but it is a zoo with some of the highest exhibit standards in Europe. Other Dutch zoos are also all worth a visit, but it really depends on how much time you have & what you want to see.
In Western Germany, Zoo Cologne is the primate hotspot, with a large an varied collection, including quite some rarities. This is a very solid allround city zoo, well worth a visit. There are plenty of other large zoos worth visiting in that area, but none that stick out for a primate lover.
So in the end it all depends on how much time you have (and mode of transport), how far you are willing to travel and what exactly you want to see. For Belgium, Netherlands & Germany Zootierliste is generally up-to-date, especially for primates, so that is a good tool.
