  We construct and study the moduli of hypersurfaces in toric orbifolds. Let
$X$ be a projective toric orbifold and $\alpha \in Cl(X)$ an ample class. The
moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G
= Aut(X)$. Since the group $G$ is non-reductive in general, we use new
techniques of non-reductive geometric invariant theory. Using the
$A$-discriminant we prove semistability for certain toric orbifolds. Further,
we show that quasismooth hypersurfaces in a weighted projective space are
stable when the weighted projective space satisfies a certain condition. We
also discuss how to proceed when this condition is not satisfied. We prove that
the automorphism group of a quasismooth hypersurface of weighted projective
space is finite excluding some low degrees.
