  Scattering amplitudes at weak coupling are highly constrained by Lorentz
invariance, locality and unitarity, and depend on model details only through
coupling constants and particle content. In this paper, we develop an
understanding of inflationary correlators which parallels that of flat-space
scattering amplitudes. Specifically, we study slow-roll inflation with weak
couplings to extra massive particles, for which all correlators are controlled
by an approximate conformal symmetry on the boundary of the spacetime. After
classifying all possible contact terms in de Sitter space, we derive an
analytic expression for the four-point function of conformally coupled scalars
mediated by the tree-level exchange of massive scalars. Conformal symmetry
implies that the correlator satisfies a pair of differential equations with
respect to spatial momenta, encoding bulk time evolution in purely boundary
terms. The absence of unphysical singularities completely fixes this
correlator. A spin-raising operator relates it to the correlators associated
with the exchange of particles with spin, while weight-shifting operators map
it to the four-point function of massless scalars. We explain how these de
Sitter four-point functions can be perturbed to obtain inflationary three-point
functions. We reproduce many classic results in the literature and provide a
complete classification of all inflationary three- and four-point functions
arising from weakly broken conformal symmetry. The inflationary bispectrum
associated with the exchange of particles with arbitrary spin is completely
characterized by the soft limit of the simplest scalar-exchange four-point
function of conformally coupled scalars and a series of contact terms. Finally,
we demonstrate that the inflationary correlators contain flat-space scattering
amplitudes via a suitable analytic continuation of the external momenta.
