Children of Ved Vyaas
After the birth of Ved Vyaas, when Satyavatee (Vyaas Jee's mother) was married to Hastinaapur King Shaantanu. She had two sons from him: Chitraangad and Vichitraveerya. When King Shaantanu died, his both children were very young. Bheeshm took care of them until they came to the age of being Raajaa. Chitraangad was elder, so he was crowned the king. He was ambitious so went out for victory over other kings. Once he started fighting with a Gandharv, named as himself (Chitraangad), just on a trivial issue that, "Why do you bear the same name as mine?". The fight lasted for three years and in the end Chitraangad was killed during that fight. Then Vichitraveerya was crowned the King of Hastinaapur. Bheeshm married him with Ambikaa and Ambaalikaa from Kaashee Pradesh. But unfortunately he also died at an early age childless.

Now Satyavatee got very worried as how to get an heir for Hastinaapur. First she requested Bheeshm, but Bheeshm was tied to his vow, so she called her first-born son, Ved Vyaas, from his Aashram. Vyaas Jee came and asked the reason to call him like this. She told him the whole story and said - "Putra, Now according to Dharm you produce sons from these widows and give heir to Hastinaapur kingdom."

Vyaas was directly coming from his Tap, so he didn't feel that his look was proper to go to those widows, so he said to his mother - "Please give me some time - one year, so that I can make myself presentable to them. I just came from my Tap as you called me." Satyavatee sadly said - "Putra, I don't have time. Do as I say."

Vyaas Jee then asked her to send her daughter-in-law to him. Satyavatee first sent Ambikaa to him. As she went in, and saw Vyaas Jee she closed her eyes with fear (some say that it was because Vyaas Jee was very ugly and foul smelled). Vyaas Jee told this to his mother that since she closed her eyes, her child will be blind. Then Satyavatee sent her other daughter-in-law Ambaalikaa to him. As she went in Vyaas room and saw him, although she did not close her eyes, but her face color turned yellow with fear. Vyaas Jee told this to his mother and said since her face color turned yellow, she would have a yellow complexioned child. He will be all right except that his health will rather be poor.

Satyavatee got very sad hearing this. She said to Vyaas - "Putra, Give me one chance more." She tried to explain to Ambikaa again to go to Vyaas Jee but she was so much scared that she did not disobey her mother-in-law but she did not go herself. She sent her maid to him. The maid considered herself very lucky and fortunate, so she went in a very happy mood. And she was all the time happy there. Vyaas Jee said to his mother that "This time your daughter-in-law sent her maid. She will have a perfect son who will be of very pious nature and knower of Dharm. He will be known on Prithvi for a very long time." After saying this Vyaas Jee went away. Satyavatee was not happy with all this but she couldn't do anything to improve the situation.

When the time came, Ambikaa gave birth to a blind son, Dhritraashtra; Ambaalikaa gave birth to a yellow complexioned son, Paandu; and the maid-servant gave birth to Vidur Jee. Thus these three sons were of Ved Vyaas Jee.

Other Children
Thus apart from (1) Dhritraashtra, (2) Paandu and (3) Vidur,
(4) Shuk Dev Jee was also Ved Vyaas Jee's son. It is said that he was born from a she-parrot that is why his name was Shuk Dev (Shuk means parrot). Skand Puraan, Naagar Khand, Ch 147 says that Shuk Dev Jee was the son of Vyaas Jee's wife Jaabaali's daughter Pinjalaa (means Vaatikaa or small garden). When Ved Vyaas Jee had written Bhaagvat Puraan, he taught it to his son Shuk Dev. Shuk Dev Jee never went into Grihasth Dharm. According to Bhaagvat Puraan, when Pareekshit (Abhimanyu and Uttaraa's son) got the Shaap of dying in 7 days time from Shringee Rishi and other Rishi and Muni suggested him to listen to Bhaagvat Puraan, Shuk Dev Jee told Bhaagvat Kathaa to him.

Note:
(1) We have not heard of any marriage of Ved Vyaas Jee anywhere, so where this Jaabaali has come from? And who is this Jaabaali? Rishi Jaabaali or somebody else?
(2) Lord Shiv was desirous of being born as the son of Ved Vyaas, so he performed a great penance. As a result, Shiv was born as Sage Shuk Dev to Vyaas Jee. Lord Vaayu also entered the body of Sage Shuk Dev. Vyaas Jee imparted all the knowledge of the Ved, Puraan and Mahaabhaarat to Shuk Dev. Shuk Dev was made responsible for teaching the Mahaabhaarat amongst Gandharv.

Maha Bharat Tatparya Niryan says that Vyaas Jee had another son named Romharshan, but other sources say that he was his very dear disciple although he was a Soot by caste that is why he was known as "Soot Jee" everywhere. Very few people know his name. Even his son Ugrashravaa is also Soot Jee and almost none knows his name too.

Disciples of Ved Vyaas
Ved Vyaas Jee had several disciples, but his main disciples were four - Pail, Vaishampaayan, Sumantu, and Jaimini. Romharshan is also supposed to be his good disciple being even a Soot. See the first Note on Ved Vyaas-1 for other details on this.