What is the recreational pastime of exploring wild cave systems called in the Americas?

We call it spelunking in NZ too. I can remember wearing a hard hat with a carbide lamp on it, jumping over chasms and walking through an underground river. Glow-worms everywhere.

Rosie-knees, The attraction is the thrill of the unknown & exploration itself; the anticipation of what is next, what will be found.

Although not in the category of wild caves, my family visited the Oregon Caves in the late 60's and it was fascinating.