The intention of the censorship act is to prevent easy access to information that could potentially harm children or that contravenes the law.
Facebook, for example, was asked by Russia's regulators to take down a page that they were concerned promoted suicide. The social network had until Sunday to comply, and did so, having decided that the page was not in the interest of general public health.
Twitter is also said to have complied with some requests to block local access to some posts.
Critics of the law say the censorship mechanisms are easy to abuse, and that the government could use them to oppose dissent.
Microsoft recently reported government requests for user data, and the figures indicated monitoring of activists in Russia.
China, well known for aggressive Internet censorship, was in the news recently when an analysis revealed the amazing speed of Chinese human censors for the first time.
[Image: Flickr user Ed Yourdon]