The nullity $\eta=\eta(G)$ of a graph $G$ is the multiplicity of the number zero in the spectrum of $G$. The chemical importance of this graph-spectrum based invariant lies in the fact, that within the H$\ddot{u}$ckel molecular orbital model, if $\eta(G)>0$ for the molecular graph $G$, then the corresponding chemical compound is highly reactive and unstable, or nonexistent. This chapter outlines both the chemically relevant aspects of $\eta$ (most of which were obtained in the 1970s and 1980s) and the general mathematical results on $\eta$ obtained recently.