Vyas was born to Sage Parashar (grandson of Sage Vashishtha) and the fisher woman Satyavati. He was named Krushn for he was dark, Dwaipayan for he was born on an island of Yamuna. His greatest work is the classification and reorganization of the Ved. Thus he came to be called as Ved Vyas. His illuminated disciples Paila, Vaisampayana, Jaimini and Sumantu presided over 4 sections of: Rg Ved, Yajur Ved, Saam Ved and Atharv Ved respectively.
Vyas also wrote Brahma Sutra, the authoritive text on the Vedant philosophy. No study of the Vedant is complete without the study of the Brahma Sutra.
Vyas is also considered as the author of the core Puraan, called Puraan Samhita at that time. He handed this Puraan to Romharshan. Romaharshan was a historian (puranik) who seems to be a popular figure, cause it is said that his recitals thrilled (harsha) the body-hair (roma) of his listeners. The core Puraan later grew into 18 main Puraans and 18 sub Puraans.
Inspite of the monumental works done by Vyas, he is known for the great itihas written by him: the Mahabharat. At the commencement of the war, Vyas decided to write the itihas of the Kurus. He started collecting information from that point. He sent Sanjay to the battlefield as a reporter. After the death of Dhruturashtra, he started writing the itihas. It took him 3 years to complete the work. I think Vyas must have written Mahabharat upto the death of Dhruturashtra, Gandhari and Kunti. He handed over MB to his son Shuka and his 4 pupils. Each one of them retold MB to different audiences.
About 60 years after the war, Vaishampayan naratted the MB to Janmejay (Abhimanyu's Grandson) during the Sarpa-Satra. I believe Vaishampayan must have written the part after Dhruturashtra's death upto the death of the Pandav and Krushn. Vaishampayan's MB also has the answers to the questions asked by Janmejay.
Vaishampayan's recital was heard by Romaharshan's son Ugrashrava (also called Sauti). For years Sauti studied MB and visited the places mentioned in MB. Almost 50 years after Sarpa-Satra, when Adhiseem Krushn (great grandson of Janmejaya) ruled Hastinapur, Ugrashrava was visiting Naimisharanya. Here he met Sage Shounak performing a 12 year sacrifice. Sage Shounak and the other sages requested Sauti to recite the MB to them. Sauti narrated MB to Sage Shounak. Sauti's MB has the answers to the questions asked by Shounak. He must have also added the information about king Parikshit, his death at the hands of Takshak, the Sarpa Satra and the part about Sage Astik.
Years later (when? at least before 100 CE) a supplement called Harivamsha was added to MB. This has the information about the Yadav dynasty and the life of Krushn. Probably this was the time when Krushn came to be regarded as an incarnation of Vishnu. Now, along with Harivamsha, MB had attained its current size of 1,00,000 shlokas.
By the custom of the ancient India, a poet would recite his work, and somebody would write it down. So when Vyas, Vaishampayan and Sauti were reciting MB, somebody must have written it. The MB that is available, is as retold by Sauti. Thus at this point one can only speculate as to which part was written by Vyas / Vaishampayan / Sauti. This fact has given a basis for debates such as the core MB written by Vyas was of only 8,800 shloka or Geeta is not Krushn's philosophy but that of Sauti's and what not!
Anyways, even Sauti's MB is not available in its original form. What is available are the copies made from older copies of MB. Unlike the Ved, the MB was not faithfully reproduced. With time, local versions of MB had evolved. The older the copy of MB, nearer it is to the original work. The earliest known edition is from pre-Gupta era. Today, numerous Dravidi, Kashmiri, Sharada, Maithili, Devnagari, Nepali and even Javanese editions of MB are available, either in their complete forms or just a few chapters.
Chrysostom, a Greek historian, wrote in 100 CE, that the Indians had a 1,00,000 shloka Illiad. Thus one can conclude that the MB had almost reached its current volume of 1,00,000 shlokas by this time and not many changes were made to the MB after that. Secondly, MB does not mention Buddhisim, thus it must have reached its present form before 500 BCE.
Few commentaries on the MB, from the past centuries have also played an important part in the study of MB. These commentaries have been especially useful for their interpretations of older Sanskrit constructs. Of these the 12th century commentary by Neekanth Chakradhar has been the basis for modern studies on MB.
Between 1933 and 1966, the Bhandarkar Oriental Research Institue of Pune, performed the gigantic work of studying all the available editions of MB. They came up with a critical edition of MB, that has only those shlok that are common to all copies. Ensuring that the extrapolations are removed. It comes as close as possible to the MB of Sauti and has been the baseline for any MB study since its completion.