In Alabama, the answer is "probably."
Under the old common law, a dog’s first bite was free – in other words, no liability attached to the dog-owner, the presumption being that the dog-owner had no way of knowing of his dog’s vicious propensities. The law has changed.
The first exception carved out was for "vicious breeds." The courts held that some breeds of dog were inherently vicious, and as to these breeds, liability attached to the first bite. Although I am not 100% certain as I write this as to which breeds are considered to be inherently vicious, I believe Pit-Bulls and Rottweiler’s are two and obviously, the argument can be made as to other breeds as well such as German Sheppard’s, Dobermans etc.
The second exception is codified at Alabama Code 3-6-1 and basically holds a dog owner liable if his dog bites a person on the dog owner’s property, provided the person has a legal right to be there and the person does not provoke the dog.
Finally, the third exception is codified at Alabama Code 3-1-5 as well as many municipal ordinances. Basically, it holds the dog owner liable for injuries inflicted by his dog "while running at large."
Accordingly, if your dog has bitten people in the past and bites again, you are liable. If your dog is a "vicious breed" and bites, you are liable. If someone legally is on your property (your guest, the postman, your children’s friends) and your dog bites (unprovoked), you are liable. If you allow your dog to leave your property and it bites a pedestrian, a cyclist or anyone else, you are liable.
Please understand that there are exceptions regarding the above. However, the bottom line is this – if you own a dog it is in your best interests to make sure that it does not attack or bite anyone less you face the prospect of liability. Of course, most responsible dog owners would agree that not only is the fear of liability a reason to make sure that their dogs don’t bite people – it also is the right thing to do. Most homeowner’s policies do provide coverage for dog bites.