  Manganese (Mn) is a key Fe-group elements, commonly employed in stellar
population and nucleosynthesis studies to explore the role of SN Ia. We have
developed a new non-local thermodynamic equilibrium (NLTE) model of Mn,
including new photo-ionisation cross-sections and new transition rates caused
by collisions with H and H- atoms. We applied the model in combination with
1-dimensional (1D) LTE model atmospheres and 3D hydrodynamical simulations of
stellar convection to quantify the impact of NLTE and convection on the line
formation. We show that the effects of NLTE are present in Mn I and, to a
lesser degree, in Mn II lines, and these increase with metallicity and with
effective temperature of a model. Employing 3D NLTE radiative transfer, we
derive new abundance of Mn in the Sun, A(Mn)=5.52 +/- 0.03 dex, consistent with
the element abundance in C I meteorites. We also apply our methods to the
analysis of three metal-poor benchmark stars. We find that 3D NLTE abundances
are significantly higher than 1D LTE. For dwarfs, the differences between 1D
NLTE and 3D NLTE abundances are typically within 0.15 dex, however, the effects
are much larger in the atmospheres of giants owing to their more vigorous
convection. We show that 3D NLTE successfully solves the ionisation and
excitation balance for the RGB star HD 122563 that cannot be achieved by 1D LTE
or 1D NLTE modelling. For HD 84937 and HD 140283, the ionisation balance is
satisfied, however, the resonance Mn I triplet lines still show somewhat lower
abundances compared to the high-excitation lines. Our results for the benchmark
stars confirm that 1D LTE modelling leads to significant systematic biases in
Mn abundances across the full wavelength range from the blue to the IR. We also
produce a list of Mn lines that are not significantly biased by 3D and can be
reliably, within the 0.1 dex uncertainty, modelled in 1D NLTE.
