Streaming giant Netflix has become the latest company to stop operating in Russia.
It said that "given the circumstances on the ground, we have decided to suspend our service in Russia".
American Express has announced it is suspending "all operations" there, while TikTok users will not be able to upload new content.
Amex said it had taken the decision in the "light of Russia's ongoing, unjustified attack on the people of Ukraine".
Visa and Mastercard have also announced they are taking similar action, as is accountancy giant PricewaterhouseCoopers.
TikTok said it was concerned about a new "fake news" law in Russia.
"We have no choice but to suspend livestreaming and new content to our video service in Russia while we review the safety implications of this law," it said in a statement.
Its in-app messaging service will not be affected.
The Kremlin has introduced a law which means that people who spread "fake" information about Russia's armed forces could be jailed for up to 15 years.
Amex said "globally issued" cards will no longer work at shops or ATMs in Russia.
Cards issued "locally in Russia by Russian banks" will stop working outside the country.
The firm is "also terminating all business operations in Belarus".
Its latest measures are in addition to "halting our relationships with banks in Russia impacted by the US and international government sanctions".
Amex admitted its decision would affect colleagues and customers in the two countries, but said one of its company values is to "do what is right".
It said in a statement: "This principle has guided us throughout this difficult crisis and will continue to do so, as we stand by our colleagues, customers, and the international community in hoping for a peaceful resolution to this crisis."
A large number of high-profile companies have distanced themselves from Russia since its invasion of Ukraine.
PayPal has closed down its services, with chief executive Dan Schulman saying the company "stands with the international community in condemning Russia's violent military aggression".
Puma and IKEA have closed stores, Apple and Microsoft have halted product sales, and Hollywood studios have postponed the release of new movies.