In this talk, we explore the structure and solution of thermal conformal field theories (CFTs) using the conformal bootstrap approach. At finite temperature, the role of crossing symmetry is played by the Kubo-Martin-Schwinger (KMS) condition, which imposes periodicity on thermal correlation functions. We present two methods for solving the resulting KMS sum rules for thermal one-point functions, assuming knowledge of the zero-temperature CFT data. The first is a numerical method that incorporates the asymptotic operator product expansion (OPE) density of heavy operators, derived using Tauberian theorems. The second is an analytical approach based on dispersion relations. We benchmark both methods against known results in free theories and two-dimensional CFTs, and subsequently apply them to O(N) models for N=1,2,3, as well as in the large-N limit.