  In this article we explore the availability of radiopure tungsten and its
potential as high density shield material for low background applications. For
compact shield designs, conventionally, lead is used. Metallic tungsten and
tungsten pseudo-alloys reach higher densities up to 19.3 gcm$^{-3}$ and do not
exhibit a significant $^{210}$Pb activity, which is a typical intrinsic
contamination in lead. Within several $\gamma$-ray screening campaigns we were
able to identify tungsten samples with activities similar or better than 1
mBqkg$^{-1}$ in $^{232}$Th, $^{40}$K, $^{60}$Co and the second part of the
$^{238}$U decay chain. In cooperation with a manufacturer we further reduced a
persisting contamination in the first part of the $^{238}$U decay chain by a
factor of $\sim$2.5 down to (305$\pm$30) mBqkg$^{-1}$. With Monte Carlo
simulations, the construction of prototype tungsten-based setups and dedicated
measurements, the shield capability of tungsten in comparison to lead was
extensively studied. Specifically, the impact of cosmogenic radiation on the
background at shallow depth was investigated. We showed that a 6-40% reduction
(depending on the exact shield configuration) in the muon-induced neutron
fluence is achievable by replacing lead with an equivalent amount of tungsten
regarding the suppression of external $\gamma$-radiation. Overall, many
benefits using tungsten especially for low energy applications below a few 100
keV are found. The pseudo-tungsten alloy presented in this work paves the way
for several applications especially regarding background suppression in
particle and astroparticle physics search programs.
