Abstract: In this talk I will describe how causality and the analytic structure of scattering amplitudes impose non-trivial constraints on the standard model effective field theory (SMEFT). In the SMEFT, there are 64 independent operators at mass dimension eight that are quartic in bosons (either Higgs or gauge fields) and that contain four derivatives and/or field strengths, including both CP-conserving and CP-violating operators. Using analytic dispersion relation arguments for two-to-two bosonic scattering amplitudes, I will describe how to derive 27 independent bounds on the sign or magnitude of the couplings. These same bounds also follow as a consequence of causality of signal propagation in nonvacuum SM backgrounds. The bounds come in two qualitative forms: i) positivity of (various linear combinations of) couplings of CP-even operators and ii) upper bounds on the magnitude of CP-odd operators in terms of (products of) CP-even couplings. I will also show how the bounds are satisfied by a wide variety of example completions. The bounds have consequences for current and future particle physics experiments, as part of the observable parameter space is inconsistent with causality and analyticity. To demonstrate the impact of the bounds, I will show how they reveal a connection between constraints derived at colliders and limits on the neutron electric dipole moment.