There are 1750 species of cacti. 1749/1750 are native only to the Americas.
The natural distribution of cactus species occurs exclusively in the New World with a single exception. This range includes North and South America, Central America, and the adjacent Islands such as the Caribbean and Galapagos islands. Rhipsalis baccifera, an epiphytic species, is the sole exception that can be found in Madagascar, tropical Africa, the Seychelles, Mauritius, Reunion, and Sri Lanka as well as tropical America.Think of all the great deserts of the world that don't have cacti: the Gobi, Sahara, Australia's Victoria. But it's worth adding this point from the Reddit discussion:
While there are not cacti in those deserts it should be noted there are cactus-like plants. Cactus belong to the family Cactaceae. These are "New World" plants that are only (except 1) endemic to the Americas. Meanwhile the Eastern continents got "Old World" plants to fill those ecological niches, many of which belong to the family Euphorbiaceae. This is not a cactus. This is also not a cactus. And another "not" cactus (image below)
The difference between cacti and cactiform succulents is explained here. You learn something every day.
Top photo credit, via the TIL subreddit.