Abstract: Let $P$ be a simple polygon with or without holes in the plane. We say two points $a$ and $b$ in $P$ are \emph{mutually visible} if the line segment $ab$ lies entirely within $P$. The {\it visibility graph} (also called the {\it vertex visibility graph}) $G$ of $P$ is defined by associating a node with each vertex of $P$ such that $(v_i, v_j)$ is an undirected edge of $G$ if polygonal vertices $v_i$ and $v_j$ are mutually visible. In this review lecture, we present open problems and conjectures on visibility graphs of polygons for graph-theoretic problems along with necessary backgrounds for understanding them.