Saiyans are a race of near-humans capable of interbreeding with Humans, successfully creating offspring. They have officially gone extinct as of 216 years after the events which took place at the conclusion of the Dragon Ball manga.
Up to the time of their demise, Saiyans developed an extremely war-based civilization. They possessed extraordinary abilities, such as the power of flight, (however not exclusive to their race) and the power to conjure energy blasts, originating from their own life force, (ki). They were eventually annexed by the Planet Trade Organization, enslaved by various factions, and commissioned to carry out the ambitious conquests of planets. They, however, managed to recieve autonomy from Lord Frieza, granting the monarch, King Vegeta, power over his people.
The beginning of the extinction of the Saiyan race began when Lord Frieza, fearful for the security of his iron rule, annihilated the Saiyans and their home planet, Vegeta. Against all odds, four Saiyans managed to survive the devestation under varying circumstances. Kakarot, more famously known as Goku, fled the planet as an infant in an advanced interplanetary space pod, along with his older brother, Raditz. Prince Vegeta, heir to the Saiyan throne was off-planet when disaster strook, and survived, along with his chaperone, Nappa. Ironically, Prince Vegeta, Nappa, and Raditz all grew up to serve Lord Frieza, quelled by the elaborate ruse that Planet Vegeta was merely destroyed by an huge meteorite.
During the events of Akira Toriyama's Dragon Ball Z saga, Nappa and Raditz were killed, leaving Vegeta and his rival Goku the remaining Saiyans. Although they both bred with Human females, (Chi Chi and Bulma) leaving powerful, half-Saiyan, and half-Human legacies in their wake, they both eventually died, ending the continuation of full-blooded Saiyans.