The first and second multiplicative Zagreb indices of a graph $G$ are $\Pi_1(G)=\sum_{x\in V(G)}d(x)^2$ and $\Pi_2(G)=\sum_{(x,y)\in E(G)}d(x)d(y)$, respectively, where $d(x)$ is the degree of the vertex $x$. We provide lower and upper bounds for $\Pi_1$ and $\Pi_2$ of a connected graph in terms of the number of vertices, number of edges, and the ordinary, additive Zagreb indices $M_1$ and $M_2$.