  The question whether a nonlinear localized mode (discrete soliton/breather)
can be mobile in a lattice has a standard interpretation in terms of the
Peierls-Nabarro (PN) potential barrier. For the most commonly studied cases,
the PN barrier for strongly localized solutions becomes large, rendering these
essentially immobile. Several ways to improve the mobility by reducing the
PN-barrier have been proposed during the last decade, and the first part gives
a brief review of such scenarios in 1D and 2D. We then proceed to discuss two
recently discovered novel mobility scenarios. The first example is the 2D
Kagome lattice, where the existence of a highly degenerate, flat linear band
allows for a very small PN-barrier and mobility of highly localized modes in a
small-power regime. The second example is a 1D waveguide array in an active
medium with intrinsic (saturable) gain and damping, where exponentially
localized, travelling discrete dissipative solitons may exist as stable
attractors. Finally, using the framework of an extended Bose-Hubbard model, we
show that while quantum fluctuations destroy the mobility of slowly moving,
strongly localized classical modes, coherent mobility of rapidly moving states
survives even in a strongly quantum regime.
