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In this paper, we present and study a new image segmentation task, called Generalized Open-set Semantic Segmentation (GOSS). Previously, with the well-known open-set semantic segmentation (OSS), the intelligent agent only detects the unknown regions without further processing, limiting their perception of the environment. It stands to reason that a further analysis of the detected unknown pixels would be beneficial. Therefore, we propose GOSS, which unifies the abilities of two well-defined segmentation tasks, OSS and generic segmentation (GS), in a holistic way. Specifically, GOSS classifies pixels as belonging to known classes, and clusters (or groups) of pixels of unknown class are labelled as such. To evaluate this new expanded task, we further propose a metric which balances the pixel classification and clustering aspects. Moreover, we build benchmark tests on top of existing datasets and propose a simple neural architecture as a baseline, which jointly predicts pixel classification and clustering under open-set settings. Our experiments on multiple benchmarks demonstrate the effectiveness of our baseline. We believe our new GOSS task can produce an expressive image understanding for future research. Code will be made available.