In a connected graph [Formula: see text], the distance between two vertices of [Formula: see text] is the length of a shortest path between these vertices. The eccentricity of a vertex [Formula: see text] in [Formula: see text] is the largest distance between [Formula: see text] and any other vertex of [Formula: see text]. The total-eccentricity index [Formula: see text] is the sum of eccentricities of all vertices of [Formula: see text]. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.