The rules mean people using Facebook, Twitter and Instagram must put their names on what they post.
Users being trolled will have the power to block unverified accounts.
This could be via an option to tick a box in their settings to receive direct messages and replies only from verified accounts.
It follows our Unmask The Online Trolls campaign to rid abusers of their cloak of anonymity. Culture Secretary Nadine Dorries said the rules will mean users are able to “stop the tidal wave of hate” they face.
A number of female politicians have received death and rape threats online.
High-profile black footballers faced racist abuse during the Euros last year.
Edleen John, director of international relations at the FA, welcomed the changes.
He said: “For too long, footballers and other participants across the game have been subjected to abhorrent discriminatory abuse from those who hide behind a cloak of anonymity. This needs to stop. The measures announced by the Government are a helpful first step to put the onus and responsibility on social media companies to create a safe space for all their users, and to give people the option to control who they interact with and what they see online.”
The Online Safety Bill will require large social media firms to give users the power to control who can interact with them, including blocking all trolls.
Imran Ahmed, chief executive of the Centre for Countering Digital Hate, said: “By starving anonymous trolls, bots and scam artists of access to their prey, social media would be a less friendly environment for them.
“And if people are forced to put their names to what they post, it will make antisocial behaviour online as risky reputationally and economically as it is offline.
“This policy is a vital tool in the battle to socialise social media.”
Ms Dorries said: “Tech firms have a responsibility to stop anonymous trolls polluting their platforms.
“We have listened to calls for us to strengthen our online safety laws and are announcing new measures to put greater power in the hands of social media users themselves.
“People will now have more control over who can contact them and be able to stop the tidal wave of hate served up to them by rogue algorithms.”
But the Government said banning anonymity online entirely would harm some users who use accounts for their personal safety, such as domestic abuse victims, activists living in authoritarian countries or young people exploring their sexuality.
It is not known how they would be able to keep their names secret.