Sudan, which the US has listed as a state sponsor of terrorism since 1993, had sought unsuccessfully to dismiss the civil lawsuit on the grounds that too much time had passed between the bombing and the filing of the lawsuit in 2004.
Lawyers are seeking to prove that Sudan granted safe haven to al-Qaeda since 1991, long before the attack on the Cole in Yemen's Aden harbor in October 2000.
"I fell to the floor on my knees. It was the most devastating thing that had ever happened to me. I felt like somebody had put their hand in the inside of my body and pulled my skin out"
father of sailor killed on USS Cole
They also hope to show that the operatives were trained at camps operated by al-Qaeda within Sudan's borders and with Sudan's permission; that Sudan's military provided al-Qaeda with at least four crates of weapons and explosives for use in terrorist activities in Yemen.
They hope to show that Osama bin Laden and Sudan's government owned businesses that provided cover for the procurement of explosives, weapons and chemicals and that Sudan gave al-Qaeda diplomatic pouches to ship explosives and weapons internationally without being searched.
Lawyers for Sudan did not offer opening statements.
Louge Gunn, whose son, Cherone Louis Gunn, was killed in the attack on the Cole, testified that he had considered suicide after learning of his son's death.
Gunn, a grief counselor by profession, described how he had thrown a chair out of the window of his Washington DC office and punched holes in the wall until his fingers bled.
"I fell to the floor on my knees. It was the most devastating thing that had ever happened to me. I felt like somebody had put their hand in the inside of my body and pulled my skin out," he said.
Andrew C Hall, an attorney for the families, opened the trial with a video recreation of the attack and actual photographs of the damage.
Hall said he expects the trial to last two to three days, with testimony by six family members and one or two experts.
Lorenzo Vidino, a terrorism expert, testified that Sudan was a safe haven for "terrorists" and that hundreds of fighters from Yemen went to the African nation for training.
"Sudan, the whole country, was a perfect sanctuary," he said.