  We present the first unbiased results for the mobility $\mu$ of
one-dimensional Holstein polaron obtained by numerical analytic continuation
combined with diagrammatic and world-line Monte Carlo methods in the
thermodynamic limit. We have identified for the first time, by the
characteristic $\omega$ and $T$ dependence in the wide region of parameters,
several distinct regimes in the $\lambda-T$ plane including band conduction
region, incoherent metallic region, activated hopping region, and high
temperature saturation region. We observe for the first time that although
mobilities and mean free paths at different values of $\lambda$ differ by many
orders of magnitude at small temperatures, their values at $T$ larger than the
bandwidth become very close to each other.
