  We review the status of $f(R,T)$ cosmological models, where $T$ is the trace
of the energy momentum tensor $T^{\mu\nu}$. We start focusing on the modified
Friedmann equations for the minimally coupled gravitational Lagrangian of the
type $f(R,T)=R +\alpha e^{\beta T} + \gamma_{n} T^{n}$. We show that in such a
minimally coupled case there exists a useful constraining relation between the
effective fractionary total matter density with an arbitrary equation of state
parameter and the modified gravity parameters. With this association the
modified gravity sector can be independently constrained using estimations of
the gas mass fraction in galaxy clusters. Using cosmological background cosmic
chronometers data and demanding the universe is old enough to accommodate the
existence of Galactic globular clusters with ages of at least $\sim 14$ Gyrs we
find a narrow range of the modified gravity free parameter space in which this
class of theories remains viable for the late time cosmological evolution. This
preferred parameter space region accommodates the $\Lambda$CDM limit of
$f(R,T)$ models. We also work out the non-minimally coupled case in the
metric-affine formalism and find that there are no viable cosmologies in the
latter situation. However, when analysing the cosmological dynamics including a
radiation component, we find that this energy density interacts with the matter
field and it does not scale according to the typical behavior. We conclude
stating that $f(R,T)$ gravity is not able to provide a full cosmological
scenario and should be ruled out as a modified gravity alternative to the dark
energy phenomena.
