I'm not expecting anyone to read this. The relationship between China and Taiwan is generally talked about under the name "Cross-Strait Relations" or something to that effect. The name comes from the Taiwan Strait, which separates the island from mainland china. A note: Mainland China is technically the People's Republic of China (PRC) and Taiwan is the Republic of China (ROC). If you read other auxilliary information about this subject, you may encounter these terms. Let's start with some basic Chinese history. For about 300 years, starting in the 17th Century, China was ruled by what is referred to as the Qing/Manchu dynasty. The territory under the purview of the Qing administration included much of modern PRC and the island of Taiwan, among other things. However, by the late 19th Century, the Qing leadership had become largely ineffective, and Japan--a rising power--became embroiled in a conflict with China. We call that conflict the Sino-Japanese War. China lost, and as part of the peace agreement, China gave Taiwan to Japan. Japan took over leadership of Taiwan and held onto it until 1945, when they were defeated in WWII. During this time, Japan developed Taiwan's economy and elevated the standard of living for many of the Taiwanese people; Japan very much groomed Taiwan to be a part of Japan, and taught Taiwanese students the Japanese language and customs. However, when Japan lost WWII, they gave control of the island back to the new Republic of China, which was ruled by what is called the Kuomintang. The Kuomintang was a product of a new wave of Chinese nationalism, and they came into power in 1912, after a successful rebellion against the Qing dynasty. The Kuomintang (KMT) aspired to be a democracy, but was also fervently nationalistic. When they were given control over Taiwan, the KMT authority was especially harsh; anti-Mainland sentiment results from actions during this time period, most notably the "228 Incident" in 1947, in which the KMT massacred Taiwanese demonstrators who were angry about brutality by the KMT Tobacco Bureau (they killed an old lady who had contraband cigarettes and took her money). Back in Mainland China, the KMT, led by Chiang Kai-Shek had begun to experience some domestic difficulties that posed a greater problem than Taiwanese demonstrators: Mao Zedong, who led a party called the Communist Party of China (CPC), was vying with the KMT for control. The situation quickly moved from political maneuvering to outright civil war. By 1949, the Communists--who called their new nation the People's Republic of China--had taken control of the mainland, and the KMT was forced to retreat to Taiwan. At this point, the PRC was thwarted every time they tried to get to Taiwan, and the KMT dared not come out. They would remain in an official state of war until 1979. Despite some flare-ups, for the most part there was a tenuous peace (a cease-fire, more accurately). Obviously, trade and travel between the ROC and the PRC was nil. Most people expected that eventually, the PRC would just take Taiwan, and that would be that. Recall, though, that in the fifties, the US became increasingly agitated at the thought of Communist control of a sizeable military, political, or economic nation. The Korean War changed the Cross-Strait situation immensely. Politically, the US realized that it could not permit the Communists to take control of Taiwan, so President Truman ordered the US Navy to hang out in the Taiwan Strait and make sure nothing happened. Things stayed like this until 1978, when eventually the ROC began to allow its citizens to visit the PRC. Eventually, this led to more and more contact--though on a very gradual scale. Eventually both sides reached what is called the 1992 Consensus, in which both the PRC and ROC decided that they should just figure out how to make things work in the present, rather than decide questions about whether both sides were separate nations. The official stance of the ROC used to be that it wanted to eventually reunify with the PRC, but it had obviously issues with the framework of the PRC government. But around the time of the 1992 Consensus, the public and the government of Taiwan began to feel more strongly about independence. The Chinese government was not happy about this, and in 1996 they conducted a missile exercise intended to intimidate pro-independence parties in Taiwan. This led to a interstrait crisis and relations between Taipei and Beijing broke down. From 1998 until around 2008, the tone of the PRC shifted dramatically. They were insistent on the "one China policy," which means pretty much what it sounds like--that Taiwan and Mainland China should all be one big China. Now, they'd had this policy before, but the general idea of the 1992 Consensus meant that this policy had fallen far into the background. Hu Jintao, the president of the PRC, continued to threaten Taiwan militarily and attempted to isolate them diplomatically--essentially refusing to deal with other nations that they perceived to be validating the legitimacy of a Taiwanese government. Such actions by the PRC were due in part to the 2000 election in Taiwan of a man named Chen Shui-bian, who was not a member of the KMT (the first time that had happened). Shui-bian was very pro-independence, and he also generally disregarded the principle of the 1992 Consensus. Though in some rare cases, the ROC and PRC worked together, for the vast majority of the time neither side spoke to the other; rather, they spoke about the other. In 2008, the KMT won the election in Taiwan again, and after some negotiations with Jintao and the PRC leadership, the 1992 Consensus was agreed upon as a framework for future talks. Direct interstrait flights began again, trade picked up. The present situation, overly simplified: the PRC and the ROC will work together, but they don't necessarily trust one another. The PRC will still diplomatically attack any nation that seems to be legitimizing the ROC--Obama, for example, sold some anti-missle equipment to the ROC, and China retaliated by severing military ties with America (this was in 2010). The PRC is ready to invade Taiwan if they declare independence, develop WMDs, or if the ROC government is severely threatened by civil disorder. Now, to your questions: Why are they determined to stay separated? The PRC believes that the ROC is an illegitimate group taking control of PRC land, the ROC believes the inverse. Are they seriously considering provoking a military conflict for the title? It's unlikely that open warfare will come of the current diplomatic situation, but if circumstances were to change (see the last sentence of my last paragraph), warfare could be possible--but it would be the PRC would would attack, not the ROC. What kind of freedoms would they give up if they integrated back into the PRC? Mainly they'd lose the autonomy of being a semisovereign nation. What legitimate claim does the ROC have to all of China? Consider the history that I laid out. The KMT controls the ROC, and the KMT used to be the people that controlled all of China--present PRC included. To them, the CPC is an illegitimate group controlling parts of their land, but they don't have the military power to stop them.