It is important to know that at any time, the Public Security Bureau (PSB) of entrance and exit may cancel your visa or residence permit. Obviously this does not happen often and only if you are considered as not complying with regulations related to visa or if you are breaking the Chinese law. If the offense is serious, it may happen to be kicked out of the country under police escort, but in most situations you have within 7 days to leave the continent. Finally, if you do not have a visa, the government can grant you a tourist visa for 7 days, to allow you to organize and leave China.
The main problem is often associated with people having an X visa (work) or a Z visa (student). Indeed, it is not permitted by the authorities to leave the country once the deadline defined by the visa is exceeded. So you do not have the opportunity to stay legally in China.
The organization you were a partner with for your arrival in china is obliged to inform the PSB of your situation and to notify that you are not with them anymore. In this case you must apply for a new visa, normally a tourist visa (L).
If you do not apply for a new visa and the PSB was informed of the situation, you will not experience difficulties until you seek to leave the country. Otherwise, you may be considered to have overstayed your visa.
Even if you have a visa for China (or maybe you do not need a visa) the airline is entitled to refuse to let you get on. Indeed, airlines are punishable by governments if passengers are not eligible for entry into a country. In general, there is no problem if you have return tickets. If you do not, you may encounter difficulties. In the case of a journey to work or study in China for one year, it is understandable that you do not wish to buy a return ticket, so this may be a problem. To solve this problem you can buy a refundable or partially refundable ticket in advance, or, book hotels for the next destination.
In some situations where it might remain doubts, you have to check with your airline in advance if it accepts your situation, and try to get a written statement. Every company has its own approach to the issue, so it is worth checking the conditions of each one of them before buying your tickets.
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This kind of counter time, due to the legislation on visas in China, can be avoided through companies specializing in PEO Services for foreigners in China. This type of company will optimize the visa process, or its period, and allow you to focus on developing your business in China.