What is the topic of this review? This review outlines the history of research on epigenetic adaptations to heat exposure. The perspective taken is that adaptations reflect properties of hormesis, whereby low, repeated doses of heat induce adaptation (acclimation/acclimatization); whereas brief, life-threatening exposures can induce maladaptive responses. What advances does it highlight? The epigenetic mechanisms underlying acclimation/acclimatization comprise specific molecular programmes on histones that regulate heat shock proteins transcriptionally and protect the organism from subsequent heat exposures, even after long delays. The epigenetic signalling underlying maladaptive responses might rely, in part, on extensive changes in DNA methylation that are sustained over time and might contribute to later health challenges. Epigenetics plays a strong role in molecular adaptations to heat by producing a molecular memory of past environmental exposures. Moderate heat, over long periods of time, induces an 'adaptive' epigenetic memory, resulting in a condition of 'resilience' to future heat exposures or cross-tolerance to other forms of toxic stress. In contrast, intense, life-threatening heat exposures, such as severe heat stroke, can result in a 'maladaptive' epigenetic memory that can place an organism at risk of later health complications. These cellular memories are coded by post-translational modifications of histones on the nucleosomes and/or by changes in DNA methylation. They operate by inducing changes in the level of gene transcription and therefore phenotype. The adaptive response to heat acclimation functions, in part, by facilitating transcription of essential heat shock proteins and exhibits a biphasic short programme (maintaining DNA integrity, followed by a long-term consolidation). The latter accelerates acclimation responses after de-acclimation. Although less studied, the maladaptive responses to heat stroke appear to be coded in long-lasting changes in DNA methylation near the promoter region of genes involved with basic cell function. Whether these memories are also encoded in histone modifications is not yet known. There is considerable evidence that both adaptive and maladaptive epigenetic responses to heat can be inherited, although most evidence comes from lower organisms. Future challenges include understanding the signalling mechanisms responsible and discovering new ways to promote adaptive responses while suppressing maladaptive responses to heat, as all life forms adapt to life on a warming planet.