  Total radiative thermal neutron-capture $\gamma$-ray cross sections for the
$^{182,183,184,186}$W isotopes were measured using guided neutron beams from
the Budapest Research Reactor to induce prompt and delayed $\gamma$ rays from
elemental and isotopically-enriched tungsten targets. These cross sections were
determined from the sum of measured $\gamma$-ray cross sections feeding the
ground state from low-lying levels below a cutoff energy, E$_{\rm crit}$, where
the level scheme is completely known, and continuum $\gamma$ rays from levels
above E$_{\rm crit}$, calculated using the Monte Carlo statistical-decay code
DICEBOX. The new cross sections determined in this work for the tungsten
nuclides are: $\sigma_{0}(^{182}{\rm W}) = 20.5(14)$ b and
$\sigma_{11/2^{+}}(^{183}{\rm W}^{m}, 5.2 {\rm s}) = 0.177(18)$ b;
$\sigma_{0}(^{183}{\rm W}) = 9.37(38)$ b and $\sigma_{5^{-}}(^{184}{\rm W}^{m},
8.33 \mu{\rm s}) = 0.0247(55)$ b; $\sigma_{0}(^{184}{\rm W}) = 1.43(10)$ b and
$\sigma_{11/2^{+}}(^{185}{\rm W}^{m}, 1.67 {\rm min}) = 0.0062(16)$ b; and,
$\sigma_{0}(^{186}{\rm W}) = 33.33(62)$ b and $\sigma_{9/2^{+}}(^{187}{\rm
W}^{m}, 1.38 \mu{\rm s}) = 0.400(16)$ b. These results are consistent with
earlier measurements in the literature. The $^{186}$W cross section was also
independently confirmed from an activation measurement, following the decay of
$^{187}$W, yielding values for $\sigma_{0}(^{186}{\rm W})$ that are consistent
with our prompt $\gamma$-ray measurement. The cross-section measurements were
found to be insensitive to choice of level density or photon strength model,
and only weakly dependent on E$_{\rm crit}$. Total radiative-capture widths
calculated with DICEBOX showed much greater model dependence, however, the
recommended values could be reproduced with selected model choices. The decay
schemes for all tungsten isotopes were improved in these analyses.
