  Motivated by the newly proposal for gravity as the effect of the torsion
scalar $T$ and trace of the energy momentum tensor $\mathcal{T}$,we investigate
the cosmological reconstruction of different models of the Universe. Our aim
here is to show that how this modified gravity model, $f(T,\mathcal{T})$ is
able to reproduce different epoches of the cosmological history. We explicitly
show that $f(T,\mathcal{T})$ can be reconstructed for $\Lambda $CDM as the most
popular and consistent model. Also we study the mathematical reconstruction of
$f(T,\mathcal{T})$ for a flat cosmological background filled by two fluids
mixture. Such model describes phantom-non-phantom era as well as the purely
phantom cosmology. We extend our investigation to more cosmological models like
perfect fluid,Chaplygin gas and massless scalar field. In each case we obtain
some specific forms of $f(T,\mathcal{T})$. These families of $f(T,\mathcal{T})$
contain arbitrary function of torsion and trace of the energy momentum.

General Relativity and Quantum Cosmology