The Italian judge hearing the retrial of Amanda Knox and and Raffaele Sollecito for the murder of Meredith Kercher has ordered a new DNA test on a knife allegedly used in the killing.
The two suspects spent four years in jail for the 2007 murder, but their convictions were overturned on appeal.
That ruling was itself overturned in March by Italy's highest court.
A separate trial convicted Rudy Guede from Ivory Coast of Miss Kercher's murder. He is serving 16 years in jail.
Neither the American Amanda Knox, nor her Italian former boyfriend Raffaele Sollecito, were in court for the start of the hearing in Florence.
Ms Knox was the housemate of Miss Kercher - a 21-year-old Briton who was found in their student lodgings in Perugia with her throat slashed.'Justice for Meredith'
Miss Kercher's sister Stephanie wrote to the court to express the family's feelings, nearly six years on from the brutal killing.
"We desperately want to discover the truth," she wrote, "and find justice for Meredith."
Miss Kercher, from Coulsdon in south London, was an undergraduate at the University of Leeds and was studying on an exchange programme at the University of Perugia at the time of her death.
Prosecutors believe she died in a drug-fuelled sexual assault.
At the first session on Monday, lawyers for the two accused requested an array of new testimony and evidence be considered by the court.
The presiding judge, Alessandro Nencini, rejected most of the defence's requests, but agreed to test for DNA on a kitchen knife which the prosecution says was used in the murder.
The first appeals trial, which cleared the two suspects, rejected considering any DNA evidence from the knife, saying the trace was too small to analyse.
The judge also agreed to the prosecution's request to hear again from a jailed gangster, who has accused his brother of murdering Miss Kercher. He will testify on Friday.
The final hearing of this new trial is expected in November, paving the way for a verdict as early as December.'Everything at stake'
Both Ms Knox and Mr Sollecito maintain their innocence.
Ms Knox insists that on the night of Miss Kercher's death she was at Mr Sollecito's flat, smoking marijuana and watching a film.
Ms Knox, 26, has exercised her right to stay away from the court, and remains at home in the US city of Seattle.
Earlier this month, she said she expected to win another acquittal, but that "common sense" told her not to return to Italy.
"I was already imprisoned as an innocent person in Italy," she told America's NBC television. "I just can't relive that."
"I thought about what it would be like to live my entire life in prison and to lose everything, to lose what I've been able to come back to and rebuild.
"I think about it all the time. It's so scary. Everything is at stake.''
However, if her previous conviction were to be confirmed, Italy would be expected to request her extradition.
Mr Sollecito, 29, is currently in the Dominican Republic, but it has been reported that he intends to return to Italy to attend parts of the retrial.