There are a variety of ways to deal with this. Depending on your personality it might be best to call them out on their ways with a couple of witnesses who will vouch for you. Often, when brought to light, adult working bullies realize that putting food on the table and keeping a roof over their head is more important than feeling like king of the hill.
Trying to avoid a bully at work can be difficult when you are forced to be in the same places so I don't advise that unless you have thousands of coworkers and are allowed to run about avoiding people.
If this is not your style you might go the route of complaining to higher ups but be sure to have at least one or two suggestions for resolving the issue, whether the person should be moved to a different department, have a private discussion with higher ups and the bully, or fired altogether.
This is a tough issue that if left unchecked can lead to workplace violence, people calling out sick, or high turnover. If you go to higher ups, make sure that you bring up these facts. Do not leave this nonsense unchecked because it negatively hurts you and the company if it is. I wrote a hub about adult relational aggression, it might have more options for you.