Veterinary drugs are pharmacologically and biologically active chemical agents. At present, veterinary drugs are extensively used to prevent and treat animal diseases, to promote animal growth, and to improve the conversion rate of feed. However, the use of veterinary drugs in food-producing animals may leave residues of the parent compounds and/or their metabolites in food products resulting in harmful effects on humans. To ensure food safety, sensitive and effective analytical methods have been developing rapidly. This review describes sample extraction and cleanup methods, and different analytical techniques are used for the determination of veterinary drug residues in milk and meat. Sample extraction methods, such as solvent extraction, liquid-liquid extraction, and cleanup methods such as dispersive solid-phase extraction and immunoaffinity chromatography, were summarized. Different types of analytical methods such as microbial, immunological, biosensor, thin layer chromatography, high-performance liquid chromatography, and liquid chromatography–tandem mass spectrometry were discussed for the analysis of veterinary drug residues in animal-derived foods. Liquid chromatography–tandem mass spectrometry is the most widely used analytical technique for the determination of antibiotic drug residues. This is due to the powerful separation of LC and accurate identification of MS, and LC-MS/MS is more popular in the analysis of veterinary drug residues.