Abstract Abstractive Sentence Summarization (ASSUM) targets at grasping the core idea of the source sentence and presenting it as the summary. It is extensively studied using statistical models or neural models based on the large-scale monolingual source-summary parallel corpus. But there is no cross-lingual parallel corpus, whose source sentence language is different to the summary language, to directly train a cross-lingual ASSUM system. We propose to solve this zero-shot problem by using resource-rich monolingual ASSUM system to teach zero-shot cross-lingual ASSUM system on both summary word generation and attention. This teaching process is along with a back-translation process which simulates source-summary pairs. Experiments on cross-lingual ASSUM task show that our proposed method is significantly better than pipeline baselines and previous works, and greatly enhances the cross-lingual performances closer to the monolingual performances. We release the code and data at https://github.com/KelleyYin/ Cross-lingual-Summarization.