Millions of women gathered in Washington and in cities around the country and the world to mount a roaring rejoinder to the inauguration of President Donald Trump.

What started as a Facebook post by a Hawaii retiree became an unprecdented international rebuke of a new president that packed cities large and small.

The Washington organisers, who originally sought a permit for 200,000, said that as many as a half million people participated in the city.

Many said they were inspired to join because of Trump's divisive campaign and his disparagement of women, minorities and immigrants. In signs and shouts, they mocked what they characterised as Trump's lewd language and sexist demeanour.

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The marches provided a balm for those eager to immerse themselves in a sea of citizens who shared their anxiety and disappointment after Democrat Hillary Clinton's historic bid for the presidency ended in defeat.

"We just want to make sure that we're heard," said Mona Osuchukwu, 27, at the Washington march with her 3-year-old daughter Chioma. "I want her to know that she has a voice. No matter what anyone tells her, especially as a black woman in America."

Organisers listed more than 670 planned events nationwide and another 70 cities overseas.

In Chicago, the demonstration was overwhelmed by its own size, after 150,000 demonstrators swamped downtown blocks. It forced officials to curtail their planned march, although thousands of protesters still paraded around the Loop. In Boston, police estimated a gathering of 125,000. In Los Angeles, officials temporarily closed some side streets to accommodate the crowds. There were huge gatherings in New York, Miami, Denver and Seattle. In Philadelphia, marchers filled city bridges. In Lexington, Kentucky, they shut down streets. In New Orleans, they played brass instruments.

Marina Knight, 43, and her 9-year-old daughter were two of the tens of thousands marching in London. "This is her first march," Knight said. "It's the first time we felt it was vital to march. I feel the rights we take for granted could go backwards, and we owe it to our daughters and the next generation to fix this somehow."

Common to every gathering in the US was fiery rhetoric, pink knit hats, and repeated references to the boast that offended so many women: Trump's infamous taped comments about groping female genitals.

Protesters got at close as they could to the presidential mansion, crowding metal barriers less than a block away.