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Query suggestion aims to suggest relevant queries for a given query, which helps users better specify their information needs. Previous work on query suggestion has been limited to the same language. In this article, we extend it to cross-lingual query suggestion (CLQS): for a query in one language, we suggest similar or relevant queries in other languages. This is very important to the scenarios of cross-language information retrieval (CLIR) and other related cross-lingual applications. Instead of relying on existing query translation technologies for CLQS, we present an effective means to map the input query of one language to queries of the other language in the query log. Important monolingual and cross-lingual information such as word translation relations and word co-occurrence statistics, and so on, are used to estimate the cross-lingual query similarity with a discriminative model. Benchmarks show that the resulting CLQS system significantly outperforms a baseline system that uses dictionary-based query translation. Besides, we evaluate CLQS with French-English and Chinese-English CLIR tasks on TREC-6 and NTCIR-4 collections, respectively. The CLIR experiments using typical retrieval models demonstrate that the CLQS-based approach has significantly higher effectiveness than several traditional query translation methods. We find that when combined with pseudo-relevance feedback, the effectiveness of CLIR using CLQS is enhanced for different pairs of languages.