  In this paper, we consider wormhole geometries in the context of teleparallel
equivalent of general relativity (TEGR) as well as $f(T)$ gravity. The TEGR is
an alternative geometrical formulation of Einstein's general relativity, where
modified teleparallel gravity or $f(T)$ gravity has been invoked as an
alternative approach for explaining an accelerated expansion of the universe.
We present the analytical solutions under the assumption of spherical symmetry
and the existence of a conformal Killing vectors to proceed a more systematic
approach in searching for exact wormhole solutions. More preciously, the
existence of a conformal symmetry places restrictions on the model. Considering
the field equations with a diagonal tetrad and anisotropic distribution of the
fluid, we study the properties of traversable wormholes in TEGR that violates
the weak and the null energy conditions at the throat and its vicinity. In the
second part, wormhole solutions are constructed in the framework of $f(T)$
gravity, where $T$ represents torsion scalar. As a consistency check, we also
discuss the behavior of energy conditions with a viable power-law $f(T)$ model
and the corresponding shape functions. In addition, a wide variety of solutions
are deduced by considering a linear equation of state relating the density and
pressure, for the isotropic and anisotropic pressure, independently of the
shape functions, and various phantom wormhole geometries are explored.
