The Xia dynasty occured before known recorded
history, and as such, there are no direct written records left by them.
Knowledge of them is lost in pre-history. Although traditional Chinese recountings begin off with the Xia, not much is known about them.
Cultures in China before the second millenia BCE are often characterised by archeologists according to their pottery. It is known that a culture using painted hand-coiled pottery existed in prominence in the northwest of modern China proper. They reached their peak in 3000 BCE. Again, not much is known about them.
A black pottery culture existed later, and reached prominence in about 2000 BCE, and as such, correspond loosely to the dates at which the Xia was meant to have existed. Their black pottery has been found all the way up the coast, from Manzhou (Manchuria) in the north, to Yue (above Vietnam) in the south.
The rulers that existed in this period are the stuff of legends. Kings that existed before and during Xia included Huang di, the Yellow Emperor, who introduced silk, ceramics, the bow and arrow, and other essentials of civilisation.
Archeological and recorded evidence only fully confirm the existence of the next recognised dynasty, the Shang, which began in approximately 1700 BCE.