Let $G$ be a graph with vertex sex $V(G)$ and edge set $E(G)$. Two much studied and in chemistry much applied graph invariants are the {ı first Zagreb index\/} $M_1=\sum\limits_{x \in V(G)} d(x)^2$ and the {ı second Zagreb index\/} $M_2=\sum\limits_{xy \in E(G)} d(x)\,d(y)$, where $d(x)$ is the degree of the vertex $x \in V(G)$. We analyze the way how these invariants were conceived in the 1970s and clarify some missing details.