  We initiate an approach to constraining conformal field theory (CFT) data at
finite temperature using methods inspired by the conformal bootstrap for vacuum
correlation functions. We focus on thermal one- and two-point functions of
local operators on the plane. The KMS condition for thermal two-point functions
is cast as a crossing equation. By studying the analyticity properties of
thermal two-point functions, we derive a "thermal inversion formula" whose
output is the set of thermal one-point functions for all operators appearing in
a given OPE. This involves identifying a kinematic regime which is the analog
of the Regge regime for four-point functions. We demonstrate the effectiveness
of the inversion formula by recovering the spectrum and thermal one-point
functions in mean field theory, and computing thermal one-point functions for
all higher-spin currents in the critical $O(N)$ model at leading order in
$1/N$. Furthermore, we develop a systematic perturbation theory for thermal
data in the large spin, low-twist spectrum of any CFT. We explain how the
inversion formula and KMS condition may be combined to algorithmically
constrain CFTs at finite temperature. Throughout, we draw analogies to the
bootstrap for vacuum four-point functions. Finally, we discuss future
directions for the thermal conformal bootstrap program, emphasizing
applications to various types of CFTs, including those with holographic duals.
