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In this paper, we study the problem of recommendation system where the users and items to be recommended are rich data structures with multiple entity types and with multiple sources of side-information in the form of graphs. We provide a general formulation for the problem that captures the complexities of modern real-world recommendations and generalizes many existing formulations. In our formulation, each user/document that requires a recommendation and each item or tag that is to be recommended, both are modeled by a set of static entities and a dynamic component. The relationships between entities are captured by several weighted bipartite graphs. To effectively exploit these complex interactions and learn the recommendation model, we propose MEDRES- a multiple graph-CNN based novel deep-learning architecture. MEDRES uses AL-GCN, a novel graph convolution network block, that harnesses strong representative features from the underlying graphs. Moreover, in order to capture highly heterogeneous engagement of different users with the system and constraints on the number of items to be recommended, we propose a novel ranking metric pAp@k along with a method to optimize the metric directly. We demonstrate effectiveness of our method on two benchmarks: a) citation data, b) Flickr data. In addition, we present two real-world case studies of our formulation and the MEDRES architecture. We show how our technique can be used to naturally model the message recommendation problem and the teams recommendation problem in the Microsoft Teams (MSTeams) product and demonstrate that it is 5-6% points more accurate than the production-grade models.