  The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced
by Hager before 1985. As its a natural counterpart, we introduced the concept
of generalized edge-connectivity $\lambda_k(G)$, recently. In this paper we
summarize the known results on the generalized connectivity and generalized
edge-connectivity. After an introductory section, the paper is then divided
into nine sections: the generalized (edge-)connectivity of some graph classes,
algorithms and computational complexity, sharp bounds of $\kappa_k(G)$ and
$\lambda_k(G)$, graphs with large generalized (edge-)connectivity,
Nordhaus-Gaddum-type results, graph operations, extremal problems, and some
results for random graphs and multigraphs. It also contains some conjectures
and open problems for further studies.
