Security concerns are high because of a violent Islamist insurgency simmering in the nearby North Caucasus region. Fears were heightened following two suicide bombings last week in the city of Volgograd, about 600 kilometers (400 miles) away, which killed 34 people and wounded scores.
Although there has been no claim of responsibility for the bombings, the leader of the Islamist insurgency last year called for attacks aimed at undermining the Sochi Olympics.
Tuesday was to be the start of a ban on all demonstrations in Sochi not connected with the games. President Vladimir Putin last weekend rescinded the blanket ban, but any demonstrations or marches still need to be approved by the Interior Ministry, which is in charge of the country's police force.
Putin has spent the past few days in Sochi, where he attended midnight church services ahead of Orthodox Christmas on Tuesday. Last week, he inspected preparations for the games and went skiing in the mountains near Sochi.
Dmitry Chernyshenko, who heads the Sochi 2014 Organizing Committee, said everything was ready with a month to go until the opening of the games.
"Every sport venue is fully prepared and has been thoroughly tested, new road and rail routes are ready to transport visitors and rehearsals for the opening ceremony are well under way," he said in a statement.
He said athletes, coaches and the media have all received their official accreditations and Russian volunteers and staff are waiting to greet them with the "warmest of welcomes."