Here's a breakdown:
* Smilodon: This is the most well-known saber-toothed cat, and lived in North and South America.
* Homotherium: These saber-toothed cats were found in Africa, Asia, Europe, and North America.
* Machairodus: Another widespread genus, found on continents like Africa, Asia, and Europe.
While they're often called "tigers," they are not directly related to modern tigers. They are more accurately categorized as extinct members of the Felidae family (the cat family).
So, to answer your question directly, saber-toothed cats evolved in various parts of the world over millions of years, with different species appearing in Africa, Asia, Europe, and North and South America.