What started this?
Washington re-imposed the sanctions after Mr Trump in May pulled out of a 2015 accord aimed at curbing Iran's nuclear ambitions.
Washington also says it wants to stop what it calls Tehran's "malign" activities including cyber attacks, ballistic missile tests, and support for terror groups and militias in the Middle East.
"We are working diligently to make sure we support the Iranian people and that we direct our activity towards ensuring that the Islamic Republic of Iran's malign behaviour is changed," US Secretary of State, Mike Pompeo, told Fox News Sunday.
"That's the goal, that's the mission, and that's what we will achieve on behalf of the president."
What could the impact be?
The US has been gradually re-imposing sanctions, but analysts say this latest round is by far the most significant.
More than 700 individuals, entities, vessels and aircraft are now on the sanctions list, including major banks, oil exporters and shipping companies.
Mr Pompeo has said that more than 100 big international companies had withdrawn from Iran because of the looming sanctions.
He also said Iranian oil exports had dropped by nearly one million barrels a day, choking the main source of funding for the country.
In addition, the Brussels-based Swift network for making international payments is expected to cut off links with targeted Iranian institutions, isolating Iran from the international financial system.
How have EU states reacted?
The UK, Germany and France - which are among the five countries still committed to the nuclear pact - have all objected to the sanctions.
They have promised to support European firms that do "legitimate business" with Iran and have set up an alternative payment mechanism - or Special Purpose Vehicle (SPV) - that will help companies trade without facing US penalties.
However, analysts doubt this will materially lessen the impact of sanctions on Iran.
And in recent days US Treasury Secretary Steven Mnuchin said the US would "aggressively" target any firm or organisation "evading our sanctions".
Who is exempt?
The Trump administration has granted exemptions to eight countries to continue importing Iranian oil, without naming them.
They are reported to include US allies Italy, India, Japan and South Korea, along with Turkey, China and India.
Mr Pompeo said the countries had already made "significant reductions in their crude oil exports" but needed "a little bit more time to get to zero".
He said two would eventually stop imports and the other six greatly reduce them.
What has the reaction been in Iran?
The US sanctions are timed to coincide with the siege of the US embassy on 4 November 1979 , which took place soon after the fall of the US-backed shah.
Some 52 Americans were held hostage in the embassy for 444 days and the two countries have been enemies ever since.
Hardliners hold protests to commemorate the siege every year but on Sunday, protesters also vented their fury about the sanctions.
Iranian state media said millions turned out in towns and cities, swearing allegiance to the Supreme Leader Ayatollah Ali Khamenei, although the BBC was unable to independently verify this figure.
It followed a fiery speech from Ayatollah Khamenei on Saturday, in which he warned the US would not "re-establish the domination" it had over Iran before 1979.
However, some Iranians have taken to Twitter to vent their frustration with the regime, with the hashtag #Sorry_US_Embassy_Siege attracting more than 19,000 tweets.
One user tweeted in English: "Over the past 40 years, the Islamic regime of Iran tried to present the US and Israel as Iran's enemies. But Iranian people do not think like mullahs. We love all nations and all people of the world."
Another said: "America is not our enemy, our enemies have taken us as hostages in our own home [country]."
