The effects of the climate change including an increase in the average global temperatures, and abnormal weather events such as frequent and severe heatwaves are emerging as a worldwide ecological concern due to their impacts on plant vegetation and crop productivity. In this review, the molecular processes of plants in response to heat stress—from the sensing of heat stress, the subsequent molecular cascades associated with the activation of heat shock factors and their primary targets (heat shock proteins), to the cellular responses—have been summarized with an emphasis on the classification and functions of heat shock proteins. Vegetables contain many essential vitamins, minerals, antioxidants, and fibers that provide many critical health benefits to humans. The adverse effects of heat stress on vegetable growth can be alleviated by developing vegetable crops with enhanced thermotolerance with the aid of various genetic tools. To achieve this goal, a solid understanding of the molecular and/or cellular mechanisms underlying various responses of vegetables to high temperature is imperative. Therefore, efforts to identify heat stress-responsive genes including those that code for heat shock factors and heat shock proteins, their functional roles in vegetable crops, and also their application to developing vegetables tolerant to heat stress are discussed.