This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle methods for solving convex-concave nonsmooth composite saddle-point problems and establish the iteration-complexity to find an approximate saddle-point. This paper places special emphasis on the primal-dual perspective of the proximal bundle method. In particular, we discover an interesting duality between the conditional gradient method and the cutting-plane scheme used within the proximal bundle method. Leveraging this duality, we further develop novel variants of both the conditional gradient method and the cutting-plane scheme.