Let $L$ be a sequence $(\ell_1, \ell_2, . . . , \ell_n)$ of $n$ lines in $\mathbb{R}^3$ (or $\mathbb{C}^3$). We say that a sequence $L$ is a realization of a graph $G=([n],E)$ if for every $\{i,j\}\in E$, we have $\ell_i\cap\ell_j\neq \emptyset$. In the talk I will present a combinatorial characterization of graphs $G$ that have the following property: For every generic realization $L$ of $G$, that consists of n pairwise distinct lines, the lines of $L$ are either all concurrent or all coplanar. The general statements that we obtain about lines, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes-Sharir framework, and will be explained in the talk. If time permits, I will introduce some applications of the results about lines to questions in graph rigidity theory.