  Models of static wormholes within the $f(\textit{R}, \textit{T})$ extended
theory of gravity are investigated and, in particular, the family
$f(\textit{R}, \textit{T}) = R + \lambda T$, with $T = \rho + P_{r} + 2P_{l}$
being the trace of the energy-momentum tensor. Models corresponding to
different relations for the pressure components (radial and lateral), and
several equations of state (EoS), reflecting different matter content, are
worked out explicitly. The solutions obtained for the shape functions of the
generated wormholes obey the necessary metric conditions, as manifested in
other studies in the literature. The respective energy conditions reveal the
physical nature of the wormhole models thus constructed. It is found, in
particular, that for each of those considered, the parameter space can be
divided into different regions, in which the exact wormhole solutions fulfill
the null~(NEC) and the weak energy conditions~(WEC), respectively, in terms of
the lateral pressure. Moreover, the dominant energy condition~(DEC) in terms of
both pressures is also valid, while $\rho + P_{r} + 2P_{l} = 0$. A similar
solution for the theory $P_{r} = \omega_{1} \rho + \omega_{2} \rho^{2}$ is
found numerically, where $\omega_{1}$ and $\omega_{2}$ are either constant or
functions of $r$, leading to the result that the NEC in terms of the radial
pressure is also valid. For non-constant $\omega_{i}$ models, attention is
focused on the behavior $\omega_{i} \propto r^{m}$. To finish, the question is
addressed, how $f(R) = R+\alpha R^{2}$ will affect the wormhole solutions
corresponding to fluids of the form $P_{r} = \omega_{1} \rho + \omega_{2}
\rho^{2}$, in the three cases mentioned above. Issues concerning the
nonconservation of the matter energy-momentum tensor, the stability of the
solutions obtained, and the observational possibilities for testing these
models are discussed in the last section.
