Ahhh - cats - now you are into my area. You ask for Large Cats, so I am not sure if you are interested in all wild cats or just the big ones. However, the few zoos that have good small cat collections (and very few they are) also have good big cat collections.
Le Parc Des Felins (France) - This is the Promised Land for all cat lovers. They have BY FAR the best cat collection ever assembled on planet earth. And they are in huge natural enclosures - 100% of the cats. Plus multiple subspecies - (4 different kinds of tigers, each in their own enclosure, 4 different kinds of leopards - I mean subspecies of common leopard, not counting other kinds they have like snow leopard and clouded leopard). If you are a cat lover, you MUST get to France some day. About an hour east of Paris.
The Living Desert (California, USA) - One of my favorites. Mostly good exhibits (and a couple so-so) for cheetah, leopard, jaguar, puma, serval, sand cat, arabian wild cat, bobcat, caracal.
Memphis Zoo (Tennesse, USA) - Another favorite, not just for cats but a great all around zoo - one of the most underrated in the country. A circular cat loop with natural exhibits for tiger, black jaguar, leopard, snow leopard, caracal, lion, cheetah, ocelot, puma. The tigers alternate between sumatran and indian, the latter featuring all three color morphs (orange, white, golden tabby).
Cincinnati Zoo (Ohio, USA) - Collection is not as strong as when they had the old Cat House (which reopens later this month as Night Hunters). But still very strong, two kinds of tiger (white indian and malayan), white lion, cheetah, new puma and snow leopard opening this summer, plus a few small cats in Night Hunters (I forget which ones). Maybe also bobcats in childrens zoo?
Nashville Zoo (Tennesse, USA) - Not a huge cat collection, but THE PLACE to go in the states for clouded leopard. Also an excellent tiger exhibit and an excellent puma exhibit. Eurasian lynx exhibit is a bit small, but featured (as of two years ago) a typical lynx and a white lynx together. This is reason enough for any cat fanatic to visit.
Exotic Feline Breeding Compound (California, USA) - The enclosures are not much to look at, but the collection is probably the best in the states. This is the country's leading breeder of some cats. And they are not a roadside breeding farm, they are a professional facility that only works with accredited zoos. Specialty is leopards, with the largest population of amur leopards and north china leopards in the states. Recently imported a persian leopard to pair with the one they had to try to restart that program. Also house only jaguarundis on public display in America. Geoffroys cat, serval, canadian and eurasian lynx, bobcat, puma, tiger, several jaguars (both black and typical) and more.
Project Survival Cat Haven (California, USA) - Mostly big cats, tiger, black and typical leopard, cheetah, lion, clouded leopard, bobcat, caracal, jaguar. The two jaguars were imported from Latin America (maybe Guatemala?) and recently produced cubs.