  We present an analysis of an $f(T, \mathcal{T})$ extension of the
Teleparallel Equivalent of General Relativity, where $T$ denotes the torsion
and $\mathcal{T}$ the trace of the energy-momentum tensor. This extension
includes non--minimal couplings between torsion and matter. In particular, we
construct two specific models that recover the usual continuity equation,
namely, $f(T, \mathcal{T})=T+g(\mathcal{T})$ and $f(T, \mathcal{T})=T\times
g(\mathcal{T})$. We then constrain the parameters of each model by fitting the
predicted distance modulus to that measured from type Ia supernovae, and find
that both models can reproduce the late--time cosmic acceleration. We also
observe that one of the models satisfies well the observational constraints and
yields a goodness--of--fit similar to the $\Lambda$CDM model, thus
demonstrating that $f(T,\mathcal{T})$ gravity theory encompasses viable models
that can be an alternative to $\Lambda$CDM.
