There are less direct flights to Taipei today than 10 years ago. At that time you had more choices to fly directly to Taiwan without any stop in a major hub such as Hong Kong or Singapore.
Recently the situation is improving again, because of the several wisa weaver agreements that Taiwan signed with many countrie. That increased the number of visitors from and to Taiwan, so the flights too.
At the same time budget carriers have appeared and are beginning to link Taipei with different Asian capitals.
You can find more about them in this page about cheap flights to Taipei.
You can fly directly to Taipei from a few European and American airports but only with the two major Taiwanese carriers, China Airlines and Eva Air, and with the Dutch carrier KLM.
Taiwanese expats are going back home to reunite with their families. At the same time other Taiwanese take advantage of vacations to go abroad.
Flights to Taipei are fully booked and expensive 7-10 days before and after the Chinese New Year. Chinese New Year date changes each year from around the middle of January to the middle of February.
Other days to be avoided are during important trade fairs in Taipei. These periods of course change every year, you can use online services as Kayak.com to check prices and availability for a specific period of time.
Surprisingly Christmas is not a bad time for traveling because it is not a public holiday in Taiwan.
In summer a typhoon can disrupt your schedule, but normally just for a day or two.
China Airlines had a terrible safety record until the late 90's. It seems it improved a lot, both with safety and quality of the service, and now ranks 4 stars with the airlines rating system Skytrax.
I traveled several times with China Airlines between 2002 and 2005, at the beginning of my stay in Taiwan. In my experience it is a nice Asian carrier, I enjoyed traveling with them. China Airlines boasts direct connections between Taipei and:
Eva Air is an interesting choice for long flights from Europe or North America because of its "Premium Economy" seats.
"Premium Economy" tickets offer bigger seats with economy class meals and service. The ticket price is therefore between economy and business fares. Eva Air thinks (probably rightfully so) that passengers are more interested, and willing to pay for, having extra space than all the bells and whistles of a full business class. Eva Air provides flights to Taipei from the following Australian, European and American cities:
KLM does not need any introduction.
For reasons unknown to me, KLM is the only European airline to have maintained direct flights to Taipei.
KLM has a daily flight between Amsterdam Schiphol and Taoyuan International Airport. This flight makes a stop in Bangkok.
KLM is a well established airline with a good safety record and with experienced flight personnel expert. They provide a service without some "Asian" frills but friendly and substantial.
I had a good experience flying with KLM between Taipei and Bangkok. My wife flew all the way between Taipei and Amsterdam, and she was happy too with them.

There are other good reasons. If, for example,you are flying from Southern Europe (Milan or Barcelona) you will have to normally connect through one of the european major hubs: Frankfurt or London Heathrow or the usually terrible Paris CDG.
So you will waste time (because you are flying backwards many extra miles) and your experience could be not so good in those overcrowded hubs. You have higher chances to lose your luggage too (it happened two times to me in Paris).
You can instead fly directly to Singapore from Milan or Barcelona and from there take your connection to Taipei. You are going to the right direction from the beginning. Normally they will not lose your luggage in Singapore Changi Airport.
I flew with all these 3 carriers and I especially like Singapore Airlines. Their flight entertainment, so important in a long flight, is the best. The ground services are very good too.
If your stop in Singapore is long, there you can find a convenient Transit Hotel, a lot of restaurants and shops, places to relax ....
On January 2009 a return ticket to Taiwan from Europe or North America will cost you about 1,500 US dollars, in March the same ticket will cost 1,000 dollars.The best flight is not always the most expensive.
You don't need to buy tickets on the Internet, if you don't feel comfortable or you need the kind of service that only the travel agency can give you.
Anyway you can check the online services to have an estimate of prices and availability.
Once my travel agency was unable to book me a ticket to Taiwan. All the flights were fully booked since a big expo was held in Taipei. I went home, I switched on my computer and I found a deal with Expedia. I was lucky!
I don't mean that Expedia is better than the travel agency, just that airlines sell some tickets through agencies, some with the online services and some directly to their customers through their own websites and ticketing offices.
You have to check them all to find the best flight!
You can follow this plan:
Good luck with your flight to Taipei ... please come to visit again this website for updates.
Cheap Flights to Taipei - a whole page just about budget flights to Taiwan.