  We show that the well-known problem of frame dependence and violation of
local Lorentz invariance in the usual formulation of $f(T)$ gravity is a
consequence of neglecting the role of spin connection. We re-formulate $f(T)$
gravity starting, instead of the "pure-tetrad" teleparallel gravity, from the
covariant teleparallel gravity, using both the tetrad and the spin connection
as dynamical variables, resulting in the fully covariant, consistent, and
frame-independent, version of $f(T)$ gravity, which does not suffer from the
notorious problems of the usual, pure-tetrad, $f(T)$ theory. We present the
method to extract solutions for the most physically important cases, such as
the Minkowski, the FRW and the spherically-symmetric ones. We show that in the
covariant $f(T)$ gravity we are allowed to use an arbitrary tetrad in an
arbitrary coordinate system along with the corresponding spin connection,
resulting always to the same physically relevant field equations.
