  Besides the conventional hadrons containing valence quarks and valence
antiquarks, quantum chromodynamics (QCD) suggests the existence of the hybrid
hadrons containing valence gluons in addition to the quarks and antiquarks, and
some experiments may have found some. A decisive experimental confirmation of
its existence, however, is still needed. At present, lattice simulations have
offered the practicable ways of theoretically guiding us to search for the
hybrid states.
  In this dissertation, we study the spectroscopy and the decay rate of the
heavy hybrid mesons made of a heavy $b$ quark, a heavy $\bar b$ antiquark, and
a gluon ($b\bar{b}g$) to selected channels, and use lattice methods to extract
the transition matrix elements in full QCD. We are particular interested in the
spin-exotic hybrid mesons.
  For sufficiently heavy quarks (e.g., $b$ quark), we use the leading
Born-Oppenheimer (LBO) approximation to calculate the static potential energy
at all $b\bar{b}$ separations. Then, by solving the Schr\"odinger equation with
this potential, we reconstruct the motion of the heavy quarks. In a similar way
we can determine decay rates. In this dissertation, we use the numerical
lattice method to calculate the mass of the $f_0$ meson at a single lattice
spacing and light quark mass, namely, $m_{f_0} = (768 \pm 136)$ MeV. Most of
all we consider the decay channels involving the production of a scalar meson.
We obtain the partial decay rate ($\Gamma$) for the channel $ H \rightarrow
\chi_b + \pi + \pi $, namely, $ \Gamma = 3.62(98)$ MeV. All of our results are
consistent with those of other researchers. Knowledge of the masses and the
decay rates should help us considerably in experimental searches for the hybrid
mesons.
