Five accused men are attending the special fast-track court. They deny charges of rape, murder and abduction.
If convicted, they could face the death penalty. A sixth suspect is to be tried by a juvenile court.
The assault outraged India and sparked a debate about treatment of women.
The first of 80 witnesses to be produced by the prosecution takes the stand in the Saket District Court on Tuesday.
All reporting of proceedings inside the courtroom has been banned and the judge has ordered lawyers not to speak to reporters.
A key witness in the case will be the victim’s male friend who was with her when she was attacked on a bus and thrown from the vehicle.
The student died in a Singapore hospital on 29 December from her internal injuries.
She and her 28-year-old companion cannot be named for legal reasons. He arrived at the courthouse in a wheelchair, still unable to walk properly due to injuries suffered in the attack.
His father, who accompanied him into the court complex, told Agence France-Presse news agency: “My son will go to any lengths to ensure that the guilty are punished.”
The bus believed to have been used in the attack was driven into the court complex on Tuesday.
The five defendants facing trial on Tuesday are Ram Singh and his brother Mukesh, Pawan Gupta, Vinay Sharma and Akshay Thakur.
They face 13 charges, including murder, gang-rape, kidnapping and destruction of evidence.
Prosecutors say they have extensive forensic evidence, supported by the suspects’ mobile-phone records and the testimony of the dying woman and her companion.
Defence lawyers are expected to argue that the forensic evidence has been fabricated and that the rush to prosecute has led to an unsafe trial.
The trial is being closely followed in India, where the case has led to massive protests about the treatment of women and how the police and legal system tackle sex crimes.
On Sunday, Indian President Pranab Mukherjee signed a new anti-rape law which has increased the minimum sentence for those convicted of gang-rape and allows for the death penalty to be used in extreme cases.
But women’s groups have come out against the new law, saying that marital rape and sexual assaults by Indian security forces in conflict zones should have been included.
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