Wine quality and safety is affected by the food contact materials (FCMs) used. These materials are expected to protect the beverage from any chemical, physical, or biological hazard and preserve its composition stable throughout its shelf-life. However, the migration of chemical substances from FCMs is a known phenomenon and requires monitoring. This review distinguishes the migrating chemical substances to those of (i) industrial origin with potential safety effects and those of (ii) natural occurrence, principally in cork (ex. tannins) with organoleptic quality effects. The review focuses on the migration of industrial chemical contaminants. Migration testing has been applied only for cork stoppers and tops, while other materials like polyethylene terephthalate (PET) bottles with aluminum cups, paperboard cartons, stainless steel vats, and oak casks have been examined for the presence of chemical migrating substances only by wine analysis without migration testing. The dominant analytical techniques applied are gas and liquid chromatography coupled to mass spectrometry (MS) for the determination of organic compounds and Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES) and ICP-MS for elemental analysis. Targeted approaches are mostly applied, while limited non-target methodologies are reported. The identified migrating substances include authorized substances like phthalate plasticizers, monomers (bisphenol A), antioxidants (Irganox 1010), known but non-authorized substances (butylparaben), break-down products, oxidation products (nonylphenol), polyurethane adhesive by-products, oligomers, ink photoinitiators, and inorganic elements. A preliminary investigation of microplastics’ migration has also been reported. It is proposed that further research on the development of comprehensive workflows of target, suspect, and non-target analysis is required to shed more light on the chemical world of migration for the implementation of an efficient risk assessment and management of wine contact materials.