The US has blocked access to many websites and social media services, including Twitter, to its citizens from a number of Muslim-majority countries, including Iran, Iraq, Yemen and Somalia.
But it also blocked access for many US citizens to Facebook and Twitter.
And it has banned US companies from doing business in the US, citing threats to national security.
The executive order was signed by President Donald Trump on Thursday.
But some US lawmakers have been protesting, with at least two of them – Senators Jeff Flake of Arizona and Cory Booker of New Jersey – launching their own lawsuits against it.
The lawsuits were sparked by the Trump administration’s decision to ban certain individuals and companies from entering the country.
The senators also argued that the US should be a beacon of democracy and freedom, not an authoritarian state.
A number of organisations have also criticised the ban, including the National Immigration Law Center, the National Lawyers Guild, and the American Civil Liberties Union.