  We discuss direct and indirect probes of chirality-flipping couplings of the
top quark to Higgs and gauge bosons, considering both CP-conserving and
CP-violating observables, in the framework of the Standard Model effective
field theory. In our analysis we include current and prospective constraints
from collider physics, precision electroweak tests, flavor physics, and
electric dipole moments (EDMs). We find that low-energy indirect probes are
very competitive, even after accounting for long-distance uncertainties. In
particular, EDMs put constraints on the electroweak CP-violating dipole moments
of the top that are two to three orders of magnitude stronger than existing
limits. The new indirect constraint on the top EDM is given by $|d_t| < 5 \cdot
10^{-20}$ e cm at $90\%$ C.L.
