Answer by Raakhee V. Menon:
It’s a long lineage spreading over several generations from Dushyanta to the main heroes that we are familiar with in the Mahabharata, viz., the Pandavas and the Kauravas. The entire lineage has been traced in the Adi Parva Book of the Mahabharata. I have added the link to the page at the bootm of this answer for anybody who might be interested in tracing the lineage from Yayati to Shantanu.
Additionally, here is an image of the family tree from Puru to Parikshit…
Dushyanta (or Dushmanta) was one of the later generation descendants of Emperor Puru, who was the founder of the Puru Vamsha (that was later changed to Kuru Vamsha in honor of another descendant).
Dushyanta married a fair maiden named Shakuntala (who was the daughter of Sage Vishwamitra and the celestial nymph Menaka) in secret and fathered a son with her named Bharata.
Their love story, their separation, Dushyanta’s memory loss due to a curse and his subsequent acceptance of his wife and child is one of the much celebrated sub-stories of the Mahabharata.
Bharata went on to become one of the most valorous monarchs of the land and present day Bharat is coined from the area he reigned over which was known as Bharatvarsha.
Emperor Shantanu was born several generations after Emperor Bharata. And his descendants are the heroes of the Mahabharata as we know it today.