  We present an extension of $f(T)$ gravity, allowing for a general coupling of
the torsion scalar $T$ with the trace of the matter energy-momentum tensor
$\mathcal{T}$. The resulting $f(T,\mathcal{T})$ theory is a new modified
gravity, since it is different from all the existing torsion or curvature based
constructions. Applied to a cosmological framework, it leads to interesting
phenomenology. In particular, one can obtain a unified description of the
initial inflationary phase, the subsequent non-accelerating, matter-dominated
expansion, and then the transition to a late-time accelerating phase.
Additionally, the effective dark energy sector can be quintessence or
phantom-like, or exhibit the phantom-divide crossing during the evolution.
Moreover, in the far future the universe results either to a de Sitter
exponential expansion, or to eternal power-law accelerated expansions. Finally,
a detailed study of the scalar perturbations at the linear level reveals that
$f(T,\mathcal{T})$ cosmology can be free of ghosts and instabilities for a wide
class of ansatzes and model parameters.
