Amyotrophic lateral sclerosis (ALS) is a progressive, fatal neurodegenerative disease characterized by the selective death of motor neurons. While the most common form of ALS is sporadic and has no known cause, a small subset of cases is familial because of underlying genetic mutations. The best-studies example of familial ALS is that caused by mutations in the protein copper-zinc superoxide dismutase. The formation of SOD1-rich inclusions in the spinal cord is an early and prominent feature of SOD1-linked familial ALS in human patients and animal models of this disease. These inclusions have been shown to consist of SOD1-rich fibrils, suggesting that the conversion of soluble SOD1 into amyloid fibrils may play an important role in the etiology of familial ALS. SOD1 is also present in inclusions found in spinal cords of sporadic ALS patients, allowing speculations to arise regarding a possible involvement of SOD1 in the sporadic form of this disease. We here review the recent research on the significance, causes, and mechanisms of SOD1 fibril formation from a biophysical perspective.