Outliers refer to observations that do not conform to the expected patterns in high-dimensional data sets. When such outliers signify risks (e.g., in fraud detection) or opportunities (e.g., in spectrum sensing), harnessing the costs associated with the risks or missed opportunities necessitates mechanisms that can identify them effectively. Designing such mechanisms involves striking an appropriate balance between reliability and cost of sensing, as two opposing performance measures, where improving one tends to penalize the other. This article poses and analyzes outlying sequence detection in a hypothesis testing framework under different outlier recovery objectives and different degrees of knowledge about the underlying statistics of the outliers.