  We investigate the cosmological applications of Myrzakulov $F(R,T)$ gravity.
In this theory ones uses a specific but non-special connection, and thus both
curvature and torsion are dynamical fields related to gravity. We introduce a
parametrization that quantifies the deviation of curvature and torsion scalars
form their corresponding values obtained using the special Levi-Civita and
Weitzenb{\"{o}}ck connections, and we extract the cosmological field equations
following the mini-super-space procedure. Even for the simple case where the
action of the theory is linear in $R$ and $T$, we find that the Friedmann
equations contain new terms of geometrical origin, reflecting the non-special
connection. Applying the theory at late times we find that we can acquire the
thermal history of the universe, where dark energy can be quintessence-like or
phantom-like, or behave exactly as a cosmological constant and thus reproducing
$\Lambda$CDM cosmology. Furthermore, we show that these features are obtained
for other Lagrangian choices, too. Finally, early-time application leads to the
de Sitter solution, as well as to an inflationary realization with the desired
scale-factor evolution.
