Plans for an invasion were developed but never put into action due to fear of US intervention after the outbreak of the Korean War. See my previous answer here: https://www.reddit.com/r/AskHistorians/comments/e3s5bt/why_didnt_mao_invade_taiwan/f94zan6?utm_medium=android_app&utm_source=share&context=3

Even though the US has since withdrawn formal diplomatic recognition and military cooperation with Taiwan, the Taiwan Relations Act provides for a policy of strategic ambiguity: the US sells arms to Taiwan and reserves the right to intervene in defence of Taiwan if China attacks.

Thus, while there have been skirmishes (e.g. PRC shelling of Quemoy and Matsu on multiple occasions during Mao's lifetime) and while there is regular saber rattling today, so far there has been no direct attempt by the PRC to invade Taiwan.