In the context of impact on human health, nitrite/nitrate and related nitrogen species such as nitric oxide (NO) are a matter of increasing scientific controversy. An increase in the content of reactive nitrogen species may result in nitrosative stress—a deleterious process, which can be an important mediator of damage to cell structures, including lipids, membranes, proteins and DNA. Nitrates and nitrites are widespread in the environment and occur naturally in foods of plant origin as a part of the nitrogen cycle. Additionally, these compounds are used as additives to improve food quality and protect against microbial contamination and chemical changes. Some vegetables such as raw spinach, beets, celery and lettuce are considered to contain high concentrations of nitrates. Due to the high consumption of vegetables, they have been identified as the primary source of nitrates in the human diet. Processed meats are another source of nitrites in our diet because the meat industry uses nitrates/nitrites as additives in the meat curing process. Although the vast majority of consumed nitrates and nitrites come from natural vegetables and fruits rather than food additives, there is currently a great deal of consumer pressure for the production of meat products free of or with reduced quantities of these compounds. This is because, for years, the cancer risks of nitrates/nitrites have been considered, since they potentially convert into the nitrosamines that have carcinogenic effects. This has resulted in the development and rapid expansion of meat products processed with plant-derived nitrates as nitrite alternatives in meat products. On the other hand, recently, these two ions have been discussed as essential nutrients which allow nitric oxide production and thus help cardiovascular health. Thus, this manuscript reviews the main sources of dietary exposure to nitrates and nitrites, metabolism of nitrites/nitrates, and health concerns related to dietary nitrites/nitrates, with particular emphasis on the effect on nitrosative stress, the role of nitrites/nitrates in meat products and alternatives to these additives used in meat products.