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The language of the region is Cantonese. However, as anywhere in mainland China, Mandarin has been the language of education and government for decades, so it is also very widely spoken. In Shenzhen and Zhuhai, Mandarin predominates because of massive migration from other regions.
English is more commonly spoken here than in most regions of China, due partly to the influence of Hong Kong and partly to the extensive foreign trade connections in the region. However, it is still not all that widespread; you can expect reasonable English from staff in many hotels and tourist shops, but generally not from a taxi driver or the waitress in a local restaurant.
Other foreign languages — anything from Swedish to Swahili — are sometimes heard; this is a very cosmopolitan region. However, none are widespread or very useful for travellers.