  We present teleparallel 3D gravity and we extract circularly symmetric
solutions, showing that they coincide with the BTZ and Deser-de-Sitter
solutions of standard 3D gravity. However, extending into f(T) 3D gravity, that
is considering arbitrary functions of the torsion scalar in the action, we
obtain BTZ-like and Deser-de-Sitter-like solutions, corresponding to an
effective cosmological constant, without any requirement of the sign of the
initial cosmological constant. Finally, extending our analysis incorporating
the electromagnetic sector, we show that Maxwell-f(T) gravity accepts deformed
charged BTZ-like solutions. Interestingly enough, the deformation in this case
brings qualitatively novel terms, contrary to the pure gravitational solutions
where the deformation is expressed only through changes in the coefficients. We
investigate the singularities and the horizons of the new solutions, and
amongst others we show that the cosmic censorship can be violated. Such novel
behaviors reveal the new features that the f(T) structure brings in 3D gravity.
