  We show that despite the inherent non-locality of quantum field theories on
the Groenewold-Moyal (GM) plane, one can find a class of ${\bf C}$, ${\bf P}$,
${\bf T}$ and ${\bf CPT}$ invariant theories. In particular, these are theories
without gauge fields or with just gauge fields and no matter fields. We also
show that in the presence of gauge fields, one can have a field theory where
the Hamiltonian is ${\bf C}$ and ${\bf T}$ invariant while the $S$-matrix
violates ${\bf P}$ and ${\bf CPT}$.
  In non-abelian gauge theories with matter fields such as the electro-weak and
$QCD$ sectors of the standard model of particle physics, ${\bf C}$, ${\bf P}$,
${\bf T}$ and the product of any pair of them are broken while ${\bf CPT}$
remains intact for the case $\theta^{0i} =0$. (Here $x^{\mu} \star x^{\nu} -
x^{\nu} \star x^{\mu} = i \theta^{\mu \nu}$, $x^{\mu}$: coordinate functions,
$\theta^{\mu \nu} = -\theta^{\nu \mu}=$ constant.) When $\theta^{0i} \neq 0$,
it contributes to breaking also ${\bf P}$ and ${\bf CPT}$. It is known that the
$S$-matrix in a non-abelian theory depends on $\theta^{\mu \nu}$ only through
$\theta^{0i}$. The $S$-matrix is frame dependent. It breaks (the identity
component of the) Lorentz group. All the noncommutative effects vanish if the
scattering takes place in the center-of-mass frame, or any frame where
$\theta^{0i}P^{\textrm{in}}_{i} = 0$, but not otherwise. ${\bf P}$ and ${\bf
CPT}$ are good symmetries of the theory in this special case.
