[quote=Flooby Badoop]
Do tell. I really do have to sleep, but I'm interested.
[/quote]

The Baratheons have Targaryen blood going back to Orys, the founder/conqueror of House Baratheon. Orys was Aegon's half-brother (and a bastard). Robert had a closer link though. His father, Steffon, was King Aerys' cousin. And Steffon's mother was Rhaelle Targaryen.