Your walls are probably old cement plaster. You need to carefully remove any peeling paint and prime with an acrylic primer. Add a little water to the primer to help it penetrate any raw areas. You could also use Peel Stop by Zinsser as a first coat of primer.
Peel Stop is a latex penetrating/binding primer that will do a good job to glue together powdery areas for patching and mudding.
This is one on the most difficult surfaces to repaint due to its brittle nature. All that you can do is be careful while prepping and use the primer to help glue things together. The peeling paint is a symptom of the old plaster.