  We propose an extension of the symmetric teleparallel gravity, in which the
gravitational action $L$ is given by an arbitrary function $f$ of the
nonmetricity $Q$ and of the trace of the matter energy-momentum tensor $T$, so
that $L=f(Q,T)$. The field equations of the theory are obtained by varying the
gravitational action with respect to both metric and connection. The covariant
divergence of the field equations is obtained, with the geometry-matter
coupling leading to the nonconservation of the energy-momentum tensor. We
investigate the cosmological implications of the theory, and we obtain the
cosmological evolution equations for a flat, homogeneous and isotropic
geometry, which generalize the Friedmann equations of general relativity. We
consider several cosmological models by imposing some simple functional forms
of the function $f(Q,T)$, corresponding to additive expressions of $f(Q,T)$ of
the form $f(Q,T)=\alpha Q+\beta T$, $f(Q,T)=\alpha Q^{n+1}+\beta T$, and
$f(Q,T)=-\alpha Q-\beta T^2$. The Hubble function, the deceleration parameter,
and the matter energy density are obtained as a function of the redshift by
using analytical and numerical techniques. For all considered cases the
Universe experiences an accelerating expansion, ending with a de Sitter type
evolution. The theoretical predictions are also compared with the results of
the standard $\Lambda$CDM model.
