  We present a new three-dimensional radiative transfer (RT) code, RADAMESH,
based on a ray-tracing, photon-conserving and adaptive (in space and time)
scheme. RADAMESH uses a novel Monte Carlo approach to sample the radiation
field within the computational domain on a "cell-by-cell" basis. Thanks to this
algorithm, the computational efforts are now focused where actually needed,
i.e. within the Ionization-fronts (I-fronts). This results in an increased
accuracy level and, at the same time, a huge gain in computational speed with
respect to a "classical" Monte Carlo RT, especially when combined with an
Adaptive Mesh Refinement (AMR) scheme. Among several new features, RADAMESH is
able to adaptively refine the computational mesh in correspondence of the
I-fronts, allowing to fully resolve them within large, cosmological boxes. We
follow the propagation of ionizing radiation from an arbitrary number of
sources and from the recombination radiation produced by H and He. The chemical
state of six species (HI, HII, HeI, HeII, HeIII, e) and gas temperatures are
computed with a time-dependent, non-equilibrium chemistry solver. We present
several validating tests of the code, including the standard tests from the RT
Code Comparison Project and a new set of tests aimed at substantiating the new
characteristics of RADAMESH. Using our AMR scheme, we show that properly
resolving the I-front of a bright quasar during Reionization produces a large
increase of the predicted gas temperature within the whole HII region. Also, we
discuss how H and He recombination radiation is able to substantially change
the ionization state of both species (for the classical Stroemgren sphere test)
with respect to the widely used "on-the-spot" approximation.
