When my daughter was eight, a former friend of hers (we'll call her Brianne) suddenly turned into a bully and gave my daughter no peace. There was nothing physical, just meanness and rudeness.
I told my daughter one thing bullies hate is cheerfulness and politeness. We practiced her responses to things Brianne would say, which REALLY helped. Kind of like studying for a test, I guess. Pretending to be Brianne, I'd say something rude to my daughter and my daughter would respond by being perky and cheerful. Our two favorite and most effective responses to anything Brianne would say:
"Thanks for letting me know!" AND
"Sorry you're feeling so crabby today!"
It worked WONDERS and drove Brianne crazy enough that within two or three days she started avoiding my daughter!
Invite the bully to your home (grit your teeth) and give them a really nice time with games, tea or sleepover. Whatever it takes. Hopefully they'll want to come back. You might even find out what is making them do it.
Get onto the internet and look for sites on bullying. Childline is quite a good one. There is so much useful info on these sites.
I think when the children are young it is easier to give them solutions, by suggesting responses to bullying and practicing what to say. It is harder and more complicated when they're older.
We are dealing with a bullying issue at the moment. We have said that if our son is hit, he is to tell the aggressor that if it happens again he's going to get walloped back. It seems to have worked up to now. He has had to hit one boy back and I find it difficult to listen to him tell me as I don't want him hitting really but I don't want him hurt either.
However hard it is, try not to take over and deal with this yourself (at least in the first instance) as direct parental involvement lets the bully know that they have succeeded in their aim: to cause distress.
My daughter (now 17 and totally happy & confident) was verbally bullied when she was 11, and again at 15. We taught her to be nice to the bullies, never to "slag them off" to others, cherish the friendships she had and to grow and nurture new friendships. The bullies never knew they upset her and soon became figures to be pitied by other kids as the unkind things they said/did were in marked contrast to my daughter's generosity of spirit and friendship with others.
My younger brother has been bullied almost all through his elementary school years. He has a hard time with school. He even felt like he was bullied by teachers too. So for one year my Mum put him in a little private institute and between that, and my other brothers taking him along on activities, he has found things that he likes and is good at. He's gained a lot of confidence in himself and made more friends at school. Now he knows how to stick up for himself and though he still struggles in school, it's easier for him because the bullies don't want to bother him. They don't want to pick on someone who will stand up for themselves.