Beate Zschaepe, 38, is accused of being part of the National Socialist Underground (NSU), which killed 10 people, most of them of Turkish origin.
She denies the murder charges. Critics say the police made serious errors.
The judge later adjourned the trial for a week after the defence team accused the judge of bias.
After entering court, Ms Zschaepe stood with folded arms and turned her back on the camera.
Her lawyers lodged a legal complaint with the judge, accusing him of bias. They complained about being searched for possible weapons or other objects on arrival, while prosecutors and police were not. The judge ordered an adjournment until 14 May to consider the complaint.
The NSU case sparked controversy as police wrongly blamed the Turkish mafia before discovering the far-right cell.
The head of Germany's domestic intelligence service was eventually forced to resign over the scandal. It also emerged that intelligence files on far-right extremists were destroyed after the cell's activities came to light.
Four male defendants are also on trial with Ms Zschaepe, facing lesser charges of having helped the NSU.
She faces life in prison if convicted.
Critics have accused authorities of turning a blind eye to the crimes of right-wing extremists, the BBC's Steve Evans reports from Munich.
Officials deny this, saying mistakes occurred because the murders were spread across different regions, each with different police and security agencies.
The killings took place over a seven-year period, and none of the victims or locations was high-profile.
Ethnic Turkish community groups and anti-racism campaigners demonstrated outside the courthouse on Monday, demanding justice. Some suspect the police of institutional racism which may have helped the neo-Nazis to act with impunity, our correspondent says.
Before the trial got under way a large crowd of journalists had gathered outside, along with dozens of people hoping to get seats in the court. About 500 police officers were deployed and nearby streets were cordoned off.
Ms Zschaepe, as a founding member of the NSU, is charged with complicity in the murders of eight ethnic Turks, a  Greek immigrant and a German policewoman between 2000 and 2007.
She is also accused of involvement in 15 armed robberies, of arson, and of attempted murder via two bomb attacks.
Prosecutors say the aim of the execution-style killings was to spread fear among immigrants and prompt them to leave Germany.
Her lawyers say she is refusing to speak in court. Only the trial opening was broadcast, in line with German legal restrictions.
The four male defendants are:
The NSU cell remained undetected until Ms Zschaepe gave herself up in November 2011, after police discovered the bodies of two of her alleged accomplices.
Uwe Mundlos, 38, and Uwe Boenhardt, 34, appeared to have shot themselves after a botched bank robbery.
After their deaths, the gun used in the murders of the 10 people was discovered.
Ms Zschaepe shared a flat in Zwickau, in the old East Germany, with the two men who were found shot dead.
The arson charge against her relates to a fire which she is alleged to have started in the flat before giving herself up. She told police she was the one they were looking for.
In addition, a video emerged showing pictures of the corpses of the victims and identifying the "organisation" behind the murders as the NSU. The video had a cartoon Pink Panther totting up the number of dead.
Only then did the authorities conclude that the killings were the work of neo-Nazis.
They had previously treated some of the families of the victims as suspects in their murders.
As a result, the trial has taken on a meaning beyond the charges in court, as it is also puts the spotlight on attitudes towards the murder of members of ethnic minority groups, our correspondent says.
An earlier start date had been set for the trial, but it was delayed for weeks amid a dispute about the seat allocations, as Turkish media were not guaranteed places.
Turkish media have now been given four seats, but several leading German newspapers missed out in the lottery, AFP news agency reports.
An alleged member of a German neo-Nazi cell has gone on trial in Munich in connection with a series of racially motivated murders.