The Narumi--Katayama index of a graph $G$ denoted by $NK(G)$, is defined as $\prod\limits_{i=1}^n\deg(v_i)$. In this paper, we determine the extremal $NK(G)$ of trees, unicyclic graphs with given diameter and vertices. Moreover, the second and third minimal $NK(G)$ of unicyclic graphs with given vertices and the minimal $NK(G)$ of bicyclic graphs with given vertices are obtained.