Genghis Khan had a lot of children along the way, which has always been the prerogative of those whose destiny is heaven's will. Many of those children also had a lot of children, and many of these kids and grandkids did conquering of their own, founding dynasties in Mongolia, China, Siberia, Russia, Persia and elsewhere. These dynasties were known as the Genghisids, the descendents of Genghis Khan, and some of those lineages still endure in modern times. For instance, there was a Genghisid family of Russian aristocrats in Czarist times, one of whom was close friends with Dostoevsky. (There's a picture of the two of them in this article.)
Recent DNA research has allowed a lot of people to claim descent from Genghis Khan, but it's based on some pretty tenuous leaps of logic. At the same time, the premise is not that far fetched- the Genghisids would certainly have had far better opportunity to spread their DNA as widely as possible than just about anyone else in Central Asia at the time! The genuine Genghisid families, however- such as the Girays of Turkey- don't have to rely on a DNA test. It's all right there in the family tree.