The undesirable effects of some hazardous compounds involved in the different steps of the winemaking process may pose health risks to consumers; hence, the importance of compliance with recent international food safety standards, including the Hazard Analysis and Critical Control Point (HACCP) standards. In recent years, there has been a rise in the development of new technologies in response to the hazardous effects of chemical compounds detected during the winemaking process, whether naturally produced or added during different winemaking processes. The main purpose was to reduce the levels of some compounds, such as biogenic amines, ethyl carbamate, ochratoxin A, and sulfur dioxide. These technological advances are currently considered a necessity, because they produce wines free of health-hazardous compounds and, most importantly, help in the management and prevention of health risks. This review shows how to prevent and control the most common potential health risks of wine using a HACCP methodology.