Anthropogenic microplastics (MPs) and nanoplastics (NPs) are ubiquitous pollutants found in aquatic, food, soil and air environments. Recently, drinking water for human consumption has been considered a significant pathway for ingestion of such plastic pollutants. Most of the analytical methods developed for detection and identification of MPs have been established for particles with sizes > 10 µm, but new analytical approaches are required to identify NPs below 1 µm. This review aims to evaluate the most recent information on the release of MPs and NPs in water sources intended for human consumption, specifically tap water and commercial bottled water. The potential effects on human health of dermal exposure, inhalation, and ingestion of these particles were examined. Emerging technologies used to remove MPs and/or NPs from drinking water sources and their advantages and limitations were also assessed. The main findings showed that the MPs with sizes > 10 µm were completely removed from drinking water treatment plants (DWTPs). The smallest NP identified using pyrolysis–gas chromatography–mass spectrometry (Pyr-GC/MS) had a diameter of 58 nm. Contamination with MPs/NPs can occur during the distribution of tap water to consumers, as well as when opening and closing screw caps of bottled water or when using recycled plastic or glass bottles for drinking water. In conclusion, this comprehensive study emphasizes the importance of a unified approach to detect MPs and NPs in drinking water, as well as raising the awareness of regulators, policymakers and the public about the impact of these pollutants, which pose a human health risk.