Here's what you need to know.
The Russian sanctions will target people and entities that:
-- undermine US cybersecurity on behalf of the Russia government
-- invest certain amounts in Russia's energy export pipelines
-- conduct "significant" transactions with Russian defense and intelligence agencies (though this will come into effect six months from now)
-- commit, or assist in, serious human rights abuses
-- commit acts of "significant" corruption
-- provide support to the Syrian government to acquire arms
-- invest, or facilitate the investment of, $10 million or more in the Russian government's privatization of any state-owned asset in a one-year period that could unfairly benefit government officials or their associates.
The bill lists 12 types of sanctions that can be imposed and obliges the President to use at least five in many cases against those affected. They can include freezing assets, such as property, revoking US visas and banning exports from the United States to those sanctioned.
Why are there so many existing sanctions against Russia?
Certain Russian individuals were hit with travel bans and asset freezes, and companies were hit with restrictions on their activities in the US and EU. Russia's powerful elite, including members of President Putin's inner circle, were targeted.
The sanctions were largely aimed at hurting the Russian economy. They targeted Russian state banks and major corporations, including state oil companies, such as Rosneft, and arms makers.
Who has imposed them?
The US and EU have spearheaded sanctions against Russia, but other countries have followed suit.
Canada joined the US and EU in the initial round of specifically targeted sanctions on March 17, 2014, after a referendum in Crimea on whether people wanted to join Russia or remain part of Ukraine -- the US considered the poll to be illegitimate. Hours after those sanctions were placed, Putin signed a draft bill for the annexation of Crimea, and five days later, he officially signed that bill into law.
Other countries imposed their own sanctions in the days and months following, including Japan and Australia, as well as several non-EU countries in Europe, including Norway and Switzerland.
What impact have they had?
Measuring the impact of sanctions on Russia's economy is not an exact science, and different experts have different opinions.
But what seems clear is that the price of oil -- which Russia's economy depends on -- has had a far greater impact than sanctions.
An International Monetary Fund report in May showed that the Russian economy was climbing out of a two-year recession and was expected to grow 1.4% this year, in part on resurgent oil prices.
But the report also said that medium-term growth would be subdued, at about 1.5%, partly because of "the lingering effects of sanctions that restrain the potential to increase investment."
In 2015, when the Ukraine-related sanctions had begun to take hold, the IMF reported that the measures could shrink the economy by 9% over time.
A reduction in trade with Russia has also hit some parts of the EU, if not the economy overall, and in August 2014, Russia retaliated by placing a ban on food imports from the countries that had imposed sanctions. That hit the EU's agri-food sector profoundly.
What does Russia want?
Putin has repeatedly ridiculed the sanctions and called on countries, particularly the US, to drop them.
But Russia's economy is beginning to show new signs of life, so such calls may now be aimed at improving conditions for certain individuals, companies or sectors.
It appears that Russia has lobbied for the US to drop other sanctions imposed under the 2012 Magnitsky Act, which targets Russians whom the US considers human rights abusers.
In Putin and Trump's second meeting, at the G20 in Germany this month, Trump claims the two leaders spoke about "adoptions," an issue strongly associated with the Magnitsky sanctions. Russia banned Americans from adopting Russian children in response to the Magnitsky Act. The law was drafted after the alleged beating to death of Russian lawyer Sergei Magnitsky in prison in 2009.
Reinstating Russian adoptions would likely go hand in hand with weakening the sanctions applied under the Magnitsky Act.
Bill Browder, a key proponent behind the act, has said that Putin is desperate to have the US drop the Magnitsky sanctions, arguing that they will likely hit Putin himself in the future.
Where does the EU stand?
The EU has been at the forefront of sanctions on Russia, but it's nervous about the sanctions being mulled in US Congress.
The bill proposes sanctions on people and entities making investments of a certain level in Russian energy export pipelines. The US argues that Russia is using pipelines to exert its influence in Europe.
But the European Union says the sanctions would unfairly penalize its countries interested in their own energy security, while Germany has said the bill breaches international law.
One project in particular is stoking animosity -- the Nord Stream 2 gas pipeline project that would extend from Russia across the Baltic to Germany. It is being pursued by Russian energy giant Gazprom.
Despite efforts to reduce a reliance on energy imported from Russia, the country supplies roughly a third of the EU's natural gas.