Smilodon was discovered in Brazil in 1841 by a Danish naturalist named Peter Wilhelm Lund. Lund's cat was S. populator and this cat became world famous rather quickly. Lund, however, gave up paleontology within the next year after finding human remains fossilized with long extinct animals. Naturalists like Darwin were thrilled by this discovery while Lund took an early retirement and basically gave up science after being faced with evidence that his beliefs about the evolution of man, adapted from Cuvier's work, were basically all wrong. The history of Smilodon actually ends with S. populator and begins with S. gracilis. The latter species is thought to have evolved from Megantereon, an older saber tooth cat belonging to the same family but a different genus, in North America nearly 2.5 million years ago.
This split and subsequent evolution lasted as one North American family until probably about 1.6 million years ago when a second group split off and evolved separately from S. gracilis as it moved slowly south toward the continent of South America. This distance and need for new adaptations to cope with the new environment would have prompted the big cats to adjust or die out in the south. The cats adapted over time and became a new species within the Smilodon genus now known as S. fatalis. This species was about the size of a Siberian Tiger, the largest living cat, and was the medium build of the Smilodon family with the North American species S. gracilis being the smallest. Sometimes two subspecies are considered to belong to S. fatalis, these being S. californicus and S. floridanus.
As the cats continued to move south to more fertile hunting grounds and less competition amongst themselves and other now extinct predators such as cave lions, short faced bears, and large birds. In fact, before S. fatalis began to invade South America thanks to the connection of the Panamanian land bridge, there was not one large mega-predator of mammalian descent on the continent. The apex predator was a bird. However, when Smilodon moved in and spread out and new adaptations began to develop and a separate recognizable species evolved from S. fatalis, the birds became the targets of these predators just like the mammals were and the birds began to die out rapidly. This new species was the aforementioned S. populator. The largest and strongest of the Smilodon family, this Smilodon spread along the eastern side of the continent and at four feet tall at the shoulders, weighing almost 1,100 pounds, and possessing twelve inch long teeth, not much was going to stop its spread.
Smilodon, however, ended with S. populator as these cats are thought to have diverged in evolution from the modern felids over 14 million years ago. There are no surviving members of its family or genus and certainly not species. The cat in your house is more closely related to lions and tigers than it ever was to the saber tooth cats.