I will summarize the history of IGM temperature measurements from the Lyman-alpha forest as well as the theory for the IGM temperature after reionization. I will show that the simplest theory for thermal evolution, which has little parametric freedom, works remarkably well at reproducing recent measurements spanning 1.5 < z <4.5. This agreement allows little room for extra sources of heat beyond photoionization. At z > 5, large fluctuations in the opacity of the Lyman-alpha forest are observed on 100 Mpc scales (with some regions being completely opaque — these are the famous Gunn-Peterson troughs). I will show that these fluctuations at z~5.5 either indicate large spatial fluctuations in the temperature or in the ionizing background. For both possibilities, the amplitude of these fluctuations yield insights into the process of cosmological reionization: that it was quite extended (it temperature is the culprit) and that it ended at z~6 (for both). Finally, whether the IGM’s temperature suppresses gas accretion onto halos is NOT set by the Jeans’ mass at the cosmic mean density (or at the halo virial density) as models assume. I will describe an intuitive picture for how the pressure of the IGM (slowly) shuts off dwarf galaxy formation.