It has been known for at least 30 years that vertically narrow thermal layers form within the middle atmosphere. Two types of temperature enhancements, the low-latitude to midlatitude mesosphere inversion layer (MIL) and the high-latitude winter stratosphere temperature enhancement (STE), have both received much attention within the atmospheric science community because of their unexplained formation mechanisms and potential impacts on the middle-atmosphere global circulation. Numerous experimental, numerical, and theoretical studies have attempted to explain certain aspects of these respective thermal layers, but no one theory consistently and satisfactorily describes all the features observed. We present a review of the literature and explicitly propose a classification scheme based on the different formation mechanisms suspected to cause these events. For the MIL we demonstrate that there are two subtypes. The first one is tidally driven and tends to occur above ~85 km. This MIL originates from large-amplitude tidal waves propagating into the mesosphere and their subsequent nonlinear interactions with gravity waves, which can often create the appearance of a double MIL separated by approximately one vertical tidal wavelength (~25 km). The other subtype of MIL is formed by a climatological planetary wave dissipation mechanism that occurs at a zero-wind line. The dissipation of the planetary wave tends to generate a mesoscale (~1000 km) inversion layer in the range of 65--80 km. These two formation mechanisms explain a host of observed characteristics, including the reason behind the downward progression of some MILs and not others, the different climatological nature of the two forms of MIL events, and the relative scarcity of MIL observations at high latitudes. The STE is believed to be generated by an altogether different process, namely, the nonlinear interaction between the polar vortex and planetary waves/Aleutian High. The induced temperatures typically peak around 40 km and often exceed 300 K, generating what appears to be a low, hot stratopause. When vertical temperature profiles are combined with synoptic analyses, one observes that the STE is the consequence of high-latitude vortex interactions creating a baroclinic atmosphere, i.e., a downward adiabatic compression induced by an ageostropic flow. We summarize the details of the relationship between this feature and sudden stratospheric warmings, as well as the potential for in situ gravity wave generation. We close with a review of currently unexplained MIL/STE features and offer new directions for future middle-atmosphere thermal layer research.