Graphs are ubiquitous and ever-present data structures that have a wide range of applications involving social networks, knowledge bases and biological interactions. The evolution of a graph in such scenarios can yield important insights about the nature and activities of the underlying network, which can then be utilized for applications such as news dissemination, network monitoring, and content curation. Capturing the continuous evolution of a graph can be achieved by long-standing sub-graph queries. Although, for many applications this can only be achieved by a set of queries, state-of-the-art approaches focus on a single query scenario. In this paper, we therefore introduce the notion of continuous multi-query processing over graph streams and discuss its application to a number of use cases. To this end, we designed and developed a novel algorithmic solution for efficient multi-query evaluation against a stream of graph updates and experimentally demonstrated its applicability. Our results against two baseline approaches using real-world, as well as synthetic datasets, confirm a two orders of magnitude improvement of the proposed solution.