Tourist visas to Taiwan have an associated "maximum stay" of 30 days. For those visas, how are the days counted? Does the stay begin on the partial day that the traveler arrives or the first full day spent in the country? Does it end on the last full day spent in the country, or the actual day the traveler departs?
Most ex-pats who are in Taiwan leave the day before visa expiration. There is a lot of back and forth within the government about what means what. While one bureau will give you one awnser another will tell you a different answer. This advice was actually repeated to me by an employee at the Ministry of Foreign Affairs. I have also had issues at immigration when leaving on the last day of my visa. I got through, but they had to interview me and I almost missed my flight out of Touyuan airport.
So, in general, we say leave the day before your visa expires. If you are on a 30-day visa, that is 30-days after your arrive, although it is technically valid to day 31.
The maximum stay for a 'visa-exempt' stay in Taiwan varies: it's either 30 days or 90 days, depending upon your nationality, so it is important to check which category you fall into.
In case of Taiwan, the duration of stay is counted from day after arrival. Here's the relevant excerpt from the Taiwanese Bureau of Consular Affairs:
The duration of stay starts from the next day of arrival and is not extendable. Visa-exempt entry can not be converted to a visitor or resident visa.
End-of-stay in most countries is considered as the actual day a traveller leaves, so I assume it would be the same for Taiwan. There isn't much point in risking a half-day stay hoping the border control officers won't count it as overstaying.
Despite hearing several times that the counting of the days begins on the day after you arrive (your first full day) in Taiwan, I just found today that this is not true!
I actually didn't recall that this question and its answers is where I probably heard it first. But I also asked various more experienced people in Taipei and they agreed, but perhaps they were just giving a "default yes" without being certain.
I arrive in Taiwan on January 24 and departed today, February 24. Since January has 31 days, that means I had the full 30 days plus the mentioned "extra" day.
The woman at the check in desk told me I had to go to the immigration desk to deal with my having overstayed one day. I told her everybody had been assuring me that the counting begins the day after you arrive but she did not agree.
I went to the immigration desk and they agreed that by staying to my 31st day I had overstayed by one day.
But there was a surprise! Since today was a Monday they said it was actually OK for me to be leaving one day late for a completely different reason, that the previous day was a Sunday or holiday! I thought this was a bit odd since airports and airlines don't close on Sundays. She did not give me a slip of paper or anything else to prove I'd been OK'd to the woman at the check in desk. So that woman actually went with my passport to the immigration desk herself to double check. She was happy to have received the same answer I got.
But there was yet one more surprise when I went through Taiwan exit immigration. The man checking my passport also believed I had overstayed by one day. I told him that I'd been to the immigration desk and that it was even doublechecked. He doubted it and checked with his superior who assured him that I was in fact right (-:
Despite the link provided in another answer, in practice it seems that this is not what happens. You will waste some time getting things checked and possibly double checked, so it's probably not a good idea to check it the hard way if you arrive at the airport without much spare time.
There is another way to get one extra day after a Sunday or holiday, but plenty of airport and even immigration staff are not aware of this, so once again it will cost you extra time so is not a great idea if you arrive late at the aiport.