Networks often contain implicit structure. We introduce novel problems and methods that look for structure in networks, by grouping nodes into supernodes and edges to superedges, and then make this structure visible to the user in a smaller generalized network. This task of finding generalizations of nodes and edges is formulated as 'network summarization'. We propose models and algorithms for networks that have weights on edges, on nodes, or on both, and study three new variants of the network summarization problem. In edge-based weighted network summarization, the summarized network should preserve edge weights as well as possible. A wider class of settings is considered in path-based weighted network summarization, where the resulting summarized network should preserve longer-range connectivities between nodes. Node-based weighted network summarization in turn allows weights also on nodes and summarization aims to preserve more information related to high weight nodes. We study theoretical properties of these problems and show them to be NP-hard. We propose a range of heuristic generalization algorithms with di↵erent trade-o↵s between complexity and quality of the result. Comprehensive experiments on real data show that weighted networks can be summarized e ciently with relatively little error.