  The transverse momentum ($p_T$) distribution of Higgs bosons at hadron
colliders enables a detailed probe of its production dynamics and is a key
ingredient to precision studies of Higgs boson properties, but receives very
large QCD corrections. We obtain a precision prediction for the $p_T$ spectrum
by matching second-order (NNLO) QCD corrections at large $p_T$ with resummation
of third-order logarithmic (N$^3$LL) corrections at small $p_T$. We achieve
significantly improved results for $p_T< 35\,{\rm GeV}$ with perturbative
uncertainties $\lesssim \pm 6\%$, and thus a convergent perturbative series for
all values of $p_T$.
