Copper is an essential trace element for the human body since it is a cofactor of several enzymes and proteins and plays a pivotal role in several biological functions (e.g., respiration, protection from oxidative damage, iron metabolism, etc.), also including the central nervous system development and functioning (e.g., synthesis of neurotransmitters, myelination, activation of neuropeptides, etc.). Therefore, copper dysmetabolism is associated with different toxic effects, mainly represented by oxidative stress, and it has been reported in many neurodegenerative disorders, such as Wilson’s disease, Menkes disease, Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis. This paper shows a detailed report of how copper is involved in the pathophysiology of these diseases. Moreover, a hint on novel therapeutic approaches based on restoring copper homeostasis through metal chelators will be pointed out.