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This paper considers the problem of how to quickly and accurately determine the availability of each spectrum band for a multi-band primary system using one or few sensors. Such problem is referred to as spectrum scanning. Two cases of practical interest are studied: 1) a single sensor case in which only one spectrum band is observed at one time; and 2) a multiple sensor case in which multiple spectrum bands are observed simultaneously. For each case, scenarios with and without a scanning delay constraint are investigated. Using mathematical tools from optimal stopping theory, optimal spectrum scanning algorithms are developed to minimize a cost function that strikes a desirable trade-off between detection performance and sensing delay. In the non delay-constrained case, it is shown that the optimal scanning algorithm is a concatenated sequential probability ratio test (C-SPRT). In the delay-constrained case, the optimal scanning algorithm has a high implementation complexity and truncation algorithms are developed as alternative low complexity options. Numerical examples are provided to illustrate the effectiveness of the proposed algorithms.