The first stage of the so-called anti-tobacco ban makes it illegal to smoke on buses, trams and other municipal transport, at railway stations and airports, on lifts and bus stations, near metro and rail stations, in administrative buildings and at education and health facilities.
From June 1, 2014, the ban will be stepped up to also include ships, long-distance trains, train platforms, hotels, cafes and restaurants -- places where Russians still smoke plentifully. Cigarette advertising and sales will also be curbed significantly.
The Kremlin initiated the public health initiative last year in a bid to halve the country's smokers. An estimated 44 million Russians use nicotine on a daily basis.
Prime Minister Dmitry Medvedev said last year that smoking-related diseases kill an estimated 400,000 people in Russia every year, and that increasingly younger people are picking up the habit.
According to a survey carried out by Levada Centre last year, 81 percent of the Russians polled were in favour of banning smoking in public places.
However, doubts have been cast about the authorities' ability to enforce the law, and many people have said that the government should focus more on helping people kick the habit rather than adopting repressive measures.
Smokers have also launched a country-wide movement to fight for their rights. "The government's policies often turn the fight against smoking into the fight against smokers," the All-Russia Movement For The Rights of Smokers said on its website.
The Duma last month passed in an initial reading the bill, setting up fines for violating the law, which start from 1000 rubles ($31) for smoking in forbidden areas and go up as high as 500,000 rubles ($16,000) for illegal tobacco advertising.
In an interview with Rossiyskaya Gazeta this week the head of Moscow police Anatoly Yakunin conceded that the interior ministry has not worked out a scheme for fining people.
"There is no mechanism yet for police control of people breaching the smoking ban," he said, hoping that people will "be conscientious and start obeying the law" while police work out a way to deal with those who don't.
Several people were seen smoking by a metro station in central Moscow Saturday morning well inside the forbidden perimeter around the entrance.
According to the World Health Organisation, Russia in 2010 had the fourth most smokers of any country in the world, accounting for 4.4 percent of the world's smokers. However it was still well behind China, which accounted for 28 percent of the world's smokers.