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Existing out-of-distribution (OOD) detection literature clearly defines semantic shift as a sign of OOD but does not have a consensus over covariate shift. Samples experiencing covariate shift but not semantic shift are either excluded from the test set or treated as OOD, which contradicts the primary goal in machine learning -- being able to generalize beyond the training distribution. In this paper, we take into account both shift types and introduce full-spectrum OOD (FS-OOD) detection, a more realistic problem setting that considers both detecting semantic shift and being tolerant to covariate shift; and designs three benchmarks. These new benchmarks have a more fine-grained categorization of distributions (i.e., training ID, covariate-shifted ID, near-OOD, and far-OOD) for the purpose of more comprehensively evaluating the pros and cons of algorithms. To address the FS-OOD detection problem, we propose SEM, a simple feature-based semantics score function. SEM is mainly composed of two probability measures: one is based on high-level features containing both semantic and non-semantic information, while the other is based on low-level feature statistics only capturing non-semantic image styles. With a simple combination, the non-semantic part is cancelled out, which leaves only semantic information in SEM that can better handle FS-OOD detection. Extensive experiments on the three new benchmarks show that SEM significantly outperforms current state-of-the-art methods. Our code and benchmarks are released in https://github.com/Jingkang50/OpenOOD.