ROME — Women of all ages, and some men, took to the streets in a dozen cities around the world on Saturday, the anniversary of the 2017 Women’s March that served as a strong rebuke of President Trump’s policies.

The annual marches and rallies have taken on wider themes since then, such as challenging the rise of the far right, while also calling for an end to inequality, the gender pay gap and violence against women.

Some events were organized in response to a call from the United States to create a “Women’s Wave.” But others were held independently, and in many cases, the core message was that women’s rights are about more than Mr. Trump.

In cities across the United States, women braved subzero temperatures in some parts of the country to march even as accusations of anti-Semitism have rocked the movement and prompted questions about its future.