  Although toric orbifolds are fundamental objects in toric topology, their
cohomology rings are largely unknown except for very few special cases. The
goal of this paper is to investigate the cohomology rings of 4-dimensional
toric orbifolds. Let $X(P,\lambda)$ is a 4-dimensional toric orbifold
associated to a polygon $P$ and a characteristic function $\lambda$. Assuming
$X(P,\lambda)$ is locally smooth over a vertex of $P$, we construct an additive
basis of $H^*(X(P,\lambda);\mathbb{Z})$ and express the cup products of the
basis elements in terms of $P$ and $\lambda$. Further we derive a formula for
computing cup products in $H^*(X(P,\lambda);R)$, where $X(P,\lambda)$ is any
general 4-dimensional toric orbifold and $R$ is a principal ideal domain
satisfying a mild condition.
