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The increasingly pervasive nature of the Web, expanding to devices and things in everyday life, along with new trends in Artificial Intelligence call for new paradigms and a new look on Knowledge Representation and Processing at scale for the Semantic Web. The emerging, but still to be concretely shaped concept of "Knowledge Graphs" provides an excellent unifying metaphor for this current status of Semantic Web research. More than two decades of Semantic Web research provides a solid basis and a promising technology and standards stack to interlink data, ontologies and knowledge on the Web. However, neither are applications for Knowledge Graphs as such limited to Linked Open Data, nor are instantiations of Knowledge Graphs in enterprises - while often inspired by - limited to the core Semantic Web stack. This report documents the program and the outcomes of Dagstuhl Seminar 18371 "Knowledge Graphs: New Directions for Knowledge Representation on the Semantic Web", where a group of experts from academia and industry discussed fundamental questions around these topics for a week in early September 2018, including the following: what are knowledge graphs? Which applications do we see to emerge? Which open research questions still need be addressed and which technology gaps still need to be closed?