Drugs are used in veterinary medicine to prevent or treat animal diseases. When rationally administered to livestock following Good Veterinary Practices (GVP), they greatly contribute to improving the production of food of animal origin. Since humans can be exposed chronically to veterinary drugs through the diet, residues in food are evaluated for effects following chronic exposures. Parameters such as an acceptable daily intake (ADI), the no-observed-adverse-effect level (NOAEL), maximum residue limits (MRLs), and the withdrawal periods (WPs) are determined for each drug used in livestock. Drug residues in food exceeding the MRLs usually appear when failing the GVP application. Different factors related either to the treated animal or to the type of drug administration, and even the type of cooking can affect the level of residues in edible tissues. Residues above the MRLs can have a diverse negative impact, mainly on the consumer’s health, and favor antimicrobial resistance (AMR). Drug residue monitoring programmes are crucial to ensure that prohibited or authorized substances do not exceed MRLs. This comprehensive review article addresses different aspects of drug residues in edible tissues produced as food for human consumption and provides relevant information contributing to rational pharmacotherapy in food-producing animals.