Many employers now have a specific employment policy aimed at eliminating bullying and any form of harassment at work. Certainly, your employer should make a proper investigation into any complaint of bullying.
Your employer has a responsibility for your well-being at work (often referred to as its duty of care) and should not knowingly allow you to continue to work in a situation which is damaging to your physical or mental health.
Even where the employer is opposed to bullying, it will likely still be liable for the injuries that you suffer. This is because it is liable for the actions of its employees towards others (known as vicarious liability) provided that the bullying occurs during the course of employment.
There are provisions also making harassment by third parties on the grounds of a protected characteristic unlawful (eg a barmaid being repeatedly sexually harassed by a customer). The Conservative-led Government has pledged to withdraw this legislation.