Of all the different types of nuts, walnuts are unusual as they contain polyunsaturated fatty acids of both the omega-3 and omega-6 families, as well as monounsaturated fats, but no cholesterol
Omega-3 are generally lacking in our diets and can help reduce risk of heart disease by improving blood flow around the arteries. Walnuts are the main "non-fish" source of alpha-linolenic acid, which gets transformed into omega-3 fatty acids in our bodies.
Walnuts are also rich in antioxidants, in particular ellagic acid, which can stop the metabolic pathways that can lead to cancer and heart disease.
It does so by protecting healthy cells from free-radical damage, helping to detoxify potential cancer-causing substances, and preventing cancer cells from multiplying.
Walnuts also contain high levels of serotonin, one of the brain chemicals sometimes implicated in migraine. Further, walnut contains a dozen different pain-relieving compounds.