This review presents the most recent data on the state-of-the-art of the main compounds present in cork, their interaction with wine, and the impact that natural stoppers may have on wines’ physical-chemical and sensory properties. According to the recent scientific literature, the chemical composition of cork and the scientific relevance of the compounds extract from cork to wine over time are reviewed. Furthermore, the effect of cork compounds transfer into wines during post-bottling is also discussed, as well as their impact on the organoleptic (colour and taste) of wines. This knowledge is essential for the decision-making process undertaken by wine producers to select the stopper most suitable for their wines. In addition, sustainability is also a topic addressed since it is a natural product that generates some waste as well as the way in which this industry is adapting to the closure of the waste cycle.