I have searched the web a lot and could not find an answer that fits my situation. Here are two links that are quite contradictory:
My situation:
I've visited Portugal in December 2012 for 29 days on a Schengen visa valid from December 1st, 2012 to February 1st, 2013 for research purposes (I am a Ph.D. student). I want to visit Sweden from February 10th, 2013 to May 10th, 2013 (89 days). I have applied for a Schengen visa again (for research). How will the 90/180 rule work in my situation?
Prometheus's answer was originally correct but the Schengen Borders Code has been amended by Regulation (EU) 610/2013 to counter the court's interpretation.
As of October 18, 2013 your stay should be “no more than 90 days in any 180-day period”. In particular, you should have been present less than 90 days during the past 180 days when leaving the Schengen area. Information about this is available on the website of the the EU commission along with a calculator.
Also note that the new rules are in principle not applicable to citizens from the following countries: Antigua and Barbuda, The Bahamas, Barbados, Brazil, Saint Kitts and Nevis, Mauritius, and Seychelles. For these countries the old rules still apply because the visa waiver agreements still contain the older definition.
In any case, the rule always applies to a person. It does not matter if you have several passports, Schengen visas or nationalities.
This answer is invalid for most people as of November 2013. The old rule might still apply to countries that have a visa waiver agreement with the EU, namely Antigua and Barbuda, The Bahamas, Barbados, Brazil, Saint Kitts and Nevis, Mauritius, and Seychelles.
I got the visa today, so I am answering my own question for the benefit of others.
There indeed is a 90/180 day rule for Schengen visas. But the way the stay duration is calculated is a little different from what is seen in most forums on the web.
The 6 month (or 180 days to be precise) period starts on the day of the first entry into the Schengen zone (Note that the day of first entry means the day you physically arrive in the zone and not the day the validity of the visa starts). In that 6 month period, you can only stay in the Schengen zone for a maximum of 90 days, irrespective of whether you have a new Schengen visa issued by the same or a different Schengen country that is valid beyond this 6 month period. At the end of this 6 month period, a NEW 6 month period starts and you can again spend a maximum of 90 days in the Schengen zone, provided you have a valid visa. If your stay duration overlaps two 6 month periods, then you must individually satisfy the 90 day limit in BOTH periods. All following 6 months period will be calculated back to back from the date of the first entry, until you remain outside the Schengen zone for at least 6 months. When you stay outside for at least 6 months (continuously) and THEN enter the Schengen zone, the six month period again starts from the day of the entry. It would be as if you were entering the Schengen zone for the first time.
All this is based on page 63 of the Handbook for the processing of visa applications and the modification of issued visas, issued by the European Commission, and my experience with this visa.
This may be a good question to visit a consulate of the destination country for.
Any information that I can find on the type C Schengen Visa is that it is 90 days in any rolling 180 day period.
There is only one site that has an indication that if you have entered EU for the purpose of business or conducting a research (see Annex 12) you may be exempt from the 90 days in 180 day requirement.
However, neither in the border policies handbook nor in visa regulation documentation this particular exemption is explicitly stated.
So the more likely scenario in either the consulate or at the border is that the border control will err on the side of caution and indicate that you will violate your C visa by staying more then 90 days in 180 day period in the EU.
The official EU site now has a calculator to tell you how many days you can stay, after you input all your previous stays: https://ec.europa.eu/assets/home/visa-calculator/calculator.htm.
The UI is pretty weak, but it works.
(edit: calculator URL moved, updated)
Here is one way to think about it, based on counting days outside Schengen. Some of the corner cases seem to feel more natural when counting this way than when counting days in Schengen.
Every time you spend a day (midnight to midnight) entirely outside the Schengen area, you earn a point.
You also earn a point if you spend a day in a Schengen country that you have a residence permit or valid long-stay (type D) visa from. This also covers days where you enter or exit such a country, as long as you don't pass through other Schengen countries.
Each point expires exactly 179 days after you earn that point. In other words, a point lasts for 180 days including the day you earn it.
When you have 90 or more unexpired points, you can make use of the "short visit" rules.
Since you earn either one point a day, or none, and points last 180 days, you will never have more than 180 points.
The short-visit rules let you enter and remain in the Schengen area if
You're a citizen of a visa-free country (and traveling with your passport from that country), or
You have a valid short-stay (type C) visa (subject to its limitation of total duration and number of entries) from any Schengen country, or
You have a valid residence permit or type D visa, and are traveling (say, for business or tourism) in one or more Schengen countries other than the one that issued it.
If you haven't been to the Schengen area in the last 90 days, it's clear that you have 90 fresh points. Everything that happened before those 90 days is forever irrelevant -- at least as far as the 90/180 day rule is concerned.
Conversely, if you stay in the Schengen area for 90 days straight and leave on the last possible day, then the next day you only have 89 points left. Since you have left the area, you will now start earning new points, but your old points will expire as fast as you can earn new ones, so it will actually be 90 days until you earn a point that doesn't just replace an expiring one. Once that happens, the previous paragraph applies.
If you have come and left in the Schengen area multiple times, then you need to be more careful counting your points. This can be complex, so this is where the various automatic calculators are handy. The one break you get is that you only ever need to look 180 days back. Everything that happened more than 180 days ago is always irrelevant, as all points you earned back then will have expired by now.
There's no need to (and no point in) attempting to keep track of "when a new 180-day period begins", counting back to the first day you set foot in the Schengen area years ago. (For several months in the early 2010s the law was that you needed to do that, after the European Court of Justice ruled that this was how the old, ambiguous, text in the Borders Code had to be interpreted. But this was quickly followed by a change of the regulation to make the current system unambiguous).
There are also a some ways to enter and be legally present in the Schengen area without being on a "short visit". These are the ways you can get in legally if you don't (yet) have 90 points:
If you have a residence permit or type D visa from the Schengen country you're in.
You will earn a point a day, as described under (2) above.
If you are in transit on your way to and from the Schengen country you have a residence permit or type D visa from.
You will not earn points on transit days, but your point score will not matter since transit is not a short visit anyway.
If the country you're in happens to accept your presence based on old bilateral treaties outside the Schengen framework. This is available only for certain citizenships, only in some of the Schengen countries. In some countries, having applied for a residence permit will put you in a similar situation until you get a decision.
You will not earn points, so if you want to use this to break to 90-day barrier, you need to put this part of your visit last.
If you enjoy freedom of movement because you're an EU/EEA citizen or a qualifying family member of a citizen.
In this case the 90/180 rule doesn't apply to you, so you don't need to count points. Presumably if you lose freedom-of-movement rights, you can then act as if you had been earning points all the time -- based on a common-sense extrapolation from the rules -- but it doesn't say so in so many words.
