  We explore the existence of wormholes in the context of $f(R,T)$ gravity. The
$f(R,T)$ theory is a curvature-matter coupled modified gravity that depends on
an arbitrary function of the Ricci scalar $R$ and the trace of the
stress-energy tensor $T$. In this work, we adopt two different choices for the
matter Lagrangian density ($\mathcal{L}_m= \mathcal{P}$ and $\mathcal{L}_m=
p_r$) and investigate the impact of each one on wormhole structure. By
adequately specifying the redshift function and the shape function, we found a
variety of exact wormhole solutions in the theory. Our finding indicates that,
for both classes of wormholes the energy density is always positive throughout
the spacetime, while the radial pressure is negative. This means exotic matter
is necessary for the existence of wormholes in $f(R,T)$ gravity.
