The generalized $k$-connectivity ${\kappa }_k(G)$ of a graph $G$, introduced by Hager in 1985, is a natural generalization of the classical connectivity. As a natural counterpart, Li et al. proposed the concept of generalized $k$-edge-connectivity ${\lambda }_k(G)$. In this paper, we obtain exact values or sharp upper and lower bounds of ${\kappa }_k(G)$ and ${\lambda }_k(G)$ for join, corona and cluster graphs.