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The integrated satellite and terrestrial networks (ISTNs) is an effective solution in order to satisfy emerging requirements of high-rate multicast service and ubiquitous coverage area. In such a network, the dynamic resource allocation strategies should be investigated carefully to achieve the most optimal energy efficiency (EE) and spectral efficiency (SE). However, the resource reuse between satellite and terrestrial components may bring about more interference and other technical challenges. Consequently, achieving high-efficiency dynamic resource allocation in ISTNs is one of the most worthy research currently. In this paper, we review the recent literatures with regard to energy efficiency and spectral efficiency optimization problem and introduce related resource allocation algorithms, as well as potential future directions in this topic.