In the Peruvian Andes, these two canyons are a two-for-one deal. Both are located near the city of Arequipa, and within striking distance of Machu Picchu-bound travelers; both are more than twice as deep as the Grand Canyon.
The Colca is more famous and much more frequently visited—it’s known for its Incan and pre-Incan history and for the presence of the otherwise rare Andean Condor—but both are accessible to travelers. The town of Chivay is the local jumping-off point for Colca Canyon, but tours and local buses also depart from Arequipa, where rental cars are available too. The small rim-side village of Cabanaconde is a good spot for a short, tough hike down into the canyon itself.