The government is publishing guidelines to help schools, parents and pupils tackle the issue of "cyber-bullying".
It comes as the Anti-Bullying Alliance publishes research suggesting that up to one in five pupils has been bullied via the internet or mobile phones.
Schools minister Jim Knight said the Department for Education is to also ask technology firms to help.
Teaching union the NASUWT said teachers were also increasingly being abused via the web, e-mails and phones.
The guidelines, to be sent to schools in England, set out simple steps that schools, parents and pupils can take to prevent cyber-bullying and deal with incidents when they occur.
These include:
The Anti-Bullying Alliance research identified seven types of cyber-bullying, ranging from abusive text messages, e-mails and phone calls to bullying in internet chatrooms, social networking sites and instant messaging.
Up to one in five pupils have experienced some form of cyber-bullying according to the study, which was based on responses from 92 children aged 11-16 from 14 London schools.
Girls were significantly more likely to be subjected to cyber-bullying, especially by text message. However, around a third of victims had never told an adult about the problem.
"Unlike other forms of bullying, cyber-bullying can follow children and young people into their private spaces "
Mr Knight said: "No child should suffer the misery of bullying, online or offline, and we will support schools in tackling it in cyberspace with the same vigilance as in the playground.
"Every school should account for cyber-bullying in their compulsory anti-bullying policies, and should take firm action where it occurs."
He added that the Education and Inspections Bill would give teachers a legal right to discipline pupils.
Parents of bullies would also face court-imposed parenting orders compelling them to attend parenting classes or face £1,000 fines.
But he warned: "Unlike other forms of bullying, cyber-bullying can follow children and young people into their private spaces and outside school hours.
"This is why it is essential that parents and young people themselves should understand how to use technologies safely to protect themselves at home and outside school hours, as well as supporting their schools in dealing with incidents."
'Climate of silence'
Chairman of the Anti-Bullying Alliance, Gill Frances, added: "As our research shows, a third of those who experience cyber-bullying do not report it.
"If we are to succeed in preventing bullying, we need to break the climate of silence in which it thrives by empowering children and young people to speak out and seek help."
The NASUWT called for the guidelines to be incorporated into school discipline policies to protect staff.
"They were sending me horrible messages over MSN, and sending me e-mails through my website every day"
Its general secretary, Chris Keates, said: "In the last two years we have had cases of photographs of a teacher being superimposed on obscene images on the internet, a website established to run a hate campaign against a teacher, persistent offensive phone calls to a member of staff and e-mails being used for sexual and homophobic harassment of members.
"Cyber-bullying and harassment, whether of pupils or staff, is unacceptable and should be met with zero tolerance. It can destroy health and careers."
The Professional Association of Teachers welcomed the new guidance.
Its general secretary, Philip Parkin, said: "Bullying is not just about physically hurting others. Children and adults can be frightened or intimidated by threatening messages sent by mobiles or by e-mail, or published online.
"Cyber-bullying is an invasion of privacy from which it can be difficult to escape.
"Much cyber-bullying takes place outside school so the actions that schools can take are limited. It is therefore important that parents are involved too. "
John Carr, technology adviser for children's charity NCH said: "We know from our everyday work that any type of bullying can be a nightmare for children and young people.
"As technology has become more sophisticated, so has the way children are bullied. One in five have been bullied by mobile phone or computer and for many there is no escape."
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