Sleeping Beauty's earliest influence apparently comes from "Perceforest," a French romance first printed in 1528. While not a Sleeping Beauty tale, Perceforest (1528) contains many elements similar to the later Sleeping Beauty tales. Sone scholars debate the connection between the stories, but I believe there are enough similarities to warrant comparisons. You can read more about Perceforest on Wikipedia.
The next known version of the tale came from Giambattista Basile's "Sun, Moon, and Talia" also known more formally as Il Pentamerone, Day 5, Tale 5 (1636). This is the tale which is thought to have influenced Perrault's Sleeping Beauty, the version I have annotated on this site. Perrault included his version, the first to use Sleeping Beauty as a title, as the first tale in his Histories ou Contes du temps passé (1697).
After Perrault, the Grimms wrote down "Briar Rose" for their own collection of tales. This version is the tamest and does not involve any of the cannibalism, adultery or rape that is found in some of the earlier renditions. The Grimm version is thought to be derived from the Perrault version which preceded it, although the Grimm brothers would have vehemently denied such a connection. The Grimm's tale is the most well-known version, barring Disney's animated feature, although Perrault's title is more commonly used. The Grimm tale ends earlier than the others with Beauty awaking with the Prince's chaste kiss. The former versions like Perrault's continued the story with the marriage and the events that followed. In the earliest variations, the king or prince impregnates Beauty in her sleep and then leaves. She wakes up when she gives birth to her twin children and one suckles her finger, removing the flax.
Sleeping Beauty has been around for a long time and scholars speculate that it appears in embryonic form in a story in the Volsunga Saga. The story tells of Brynhild and her fear of being married to a cowardly man when she is banished to earth.
Perrault's story was first translated into English in Robert Samber's Histories, or Tales of Past Times (1729). A copy of that story can be found in Iona and Peter Opie's Classic Fairy Tales. Andrew Lang adapted his version from Perrault's story. Sleeping Beauty has been popular since its first publication by being printed in chapbooks and appearing in pantomimes. Perrault was also one of the last interpreters of the tale to avoid waking Beauty with a kiss. Most of the versions written and produced since then have used the kiss to awaken the sleeping princess.
Another sleeping beauty tale, "The Ninth Captain's Tale," also appears in 1,001 Arabian Nights, although its relation to the European Sleeping Beauties is tenuous at best.