This linked to list of extant megafauna is quite varied (and a little sparse?). Let's organize it a bit:
African Megafauna
- Gorilla
- Black Rhino
- Hippo
- Ostrich
- Elephant (African)
- Lion
- Giraffe
- African Buffalo
- Zebra
This is notably the largest list of any category. According to Jared Diamond in Guns, Germs, & Steel, there's a good reason for that. Humans evolved alongside these animals in Africa, so they learned to deal with the growing capabilities of Honinids slowly over millions of years, at a pace their evolution could handle, rather than getting fully-evolved Homo Sapiens dropped on them cold.
SE Asian Megafauna
- Asian Elephant
- Tiger
- Giant Boa
- Wild Water Buffalo
Jared didn't go into this one, but SE Asia was the first place we know of hominids spreading to outside of Africa (about 2 million years ago). So if his thesis holds water, then you'd expect some of these to have survived as well, since they also got introduced to hominids rather early on, when their hunting capabilities weren't as great.
Aguatic Megafauna
- Whales (toothed and untoothed)
- Crocodilians
- Sharks
- Mantas
- Assorted Giant fish
- Giant Octopi & Squid
The explanation here seems rather obvious: Humans aren't aquatic creatures. Hunting large creatures in the deep water they are capable of retreating to is no easy task for a human. Doing it at the scale required to wipe out a species seems right out prior to the modern era.
Arctic Megafauna
- Grizzly/Polar Bear
- Moose
- Musk Ox
It seems likely there's a weaker version of the effect with Aquatic animals going on here. Humans can live in the arctic, and hunt there. However, its not easy, and tougher to do at scale.
Domesticates
These survived because they were usefully domesticable. Their wild forbears are largely extinct.
Large Bovines (worldwide)
Outside of the African variety mentioned above, these creatures (barely) managed to live into the modern era, when men with horses and firearms nearly exterminated them all. Being big, fast, tough (by bow-and-arrow era standards) and relatively prolific may have helped them.
South America
This is actually a really small list, compared to what they had prior to the peopling of the continent. Anaconda are aquatic and largely live in places humans can't farm well. Rhea I don't know much about, but apparently they are very fast (40MPH). Likely the fastest land animal in South America. Their impressive survival capabilities are underscored by the fact that there's now feral populations of them in Germany and the UK.