And those who work for such organizations could be sentenced to as much as six years in prison.
The new measure, signed into law Saturday by President Vladimir Putin, has provoked an international outcry.
The U.S. State Department said it was "deeply troubled." Amnesty International said the law threatened "fundamental freedoms." Human Rights Watch called it a "piece of repressive legislation."
The law was passed earlier in the week by both houses of the Russian parliament. It says that a foreign non-governmental organization can be recognized as undesirable if it poses a threat to the constitutional order of the Russian Federation or to the country's defense and security.
The "undesirable" designation can be made by the country's prosecutor general or his deputies in consultation with the Foreign Ministry. The penalties -- both civil and criminal -- can be imposed if the organization continues to operate in Russia after having been so designated.
The law also targets Russian citizens or groups that have any "involvement" with undesirable foreign NGOs.
Supporters of the law said it was needed to stop "destructive organizations" from threatening the Russian state. But opponents portrayed it as one of a series of measures aimed at suppressing opposition and restricting freedom in the country.
'Squeezing the very life out of Russian civil society'
Tanya Lokshina, Russia program director for Human Rights Watch, said the new law had "the potential to severely damage our work in Russia," and was a cause for grave concern for all international groups operating in the country.
Nevertheless, she said she did not believe the law was aimed at international organizations like hers. Instead, she said, it was aimed at Russians who might cooperate with, or support, international organizations.
"The bill does not specify what 'involvement' might include," Lokshina wrote in an article published online. "So anything goes. Distributing -- including by posting online -- the statements, reports, or other materials of an 'undesirable,'" or participating in events or even communicating with staffers of such an organization could be cause for prosecution, she wrote.
In other words, in Lokshima's estimation, even retweeting something from an undesirable NGO could land a Russian citizen in jail.
Vague wording 'designed for selective implementation'
"The intended targets of this new legislation on foreign and international organizations are actually Russian activists and Russian groups," she wrote. "The bill is aimed at cutting them off from their international partners, further isolating them, and squeezing the very life out of Russian civil society."
Like many others, Lokshina criticized the vague wording of the bill, saying it appeared the the law was designed for selective implementation.