Amid tight security, the hearing began at 2pm (1100 GMT) on Tuesday in the presence of the media and a number of relatives. Eleven of the suspects were being tried in absentia.
Twenty-five defendants are Kuwaitis, seven are stateless Arabs know as the "bidoon", two Jordanians and one each from Saudi Arabia, Australia and Somalia, the lawyers said.
Kuwait's public prosecution has demanded the death penalty for 20 of the suspects for allegedly being members of a group linked to al-Qaida who clashed with police in January, according to Al-Qabas newspaper.
The prosecution is also seeking various jail terms for 15 other suspects in a 2000-page indictment submitted to Kuwaiti court authorities on 11 May.
The 37 are allegedly members or sympathisers of a group named the Peninsula Lions, which reportedly maintains links with the Saudi al-Haramain Brigades and Osama bin Laden's al-Qaida network.
Most of the suspects are accused of fighting four deadly gun battles with Kuwaiti security forces in January that left four police officers dead and 10 others wounded.
Eight were killed in the gunfights, while the alleged leader of the group, Amer Khlaif al-Enezi, died in police detention eight days after his arrest on 31 January.
Eight were killed in the gunfights
Al-Enezi's wife, Nuha, was the sole woman among the defendants in court, sitting in a wheelchair due to an apparent illness.
The list of defendants includes fiery Sunni Muslim cleric Sheikh Hamed al-Ali, charged with issuing fatwas, or religious edicts, for the group. He has denied the charge.
It also includes Khaled al-Dosari and Mohsen al-Fadhli who remain at large. Al-Dosari is the spokesman for the Society Against Arbitrary Detention and Torture, while Fadhli was in February added to the UN list of "terror" financiers.
The prosecution charges most of the suspects of joining an illegal group which was plotting armed confrontations with state authorities and to attack government interests.
They are also charged of murdering several police officers during gun battles with security forces.
A large number of the suspects were arrested during the clashes with the police in January last year, while the rest were arrested later, based on confessions obtained by police.