By Ray Furlong
BBC Berlin correspondent
After Sunday's inconclusive election, Germany is rife with speculation about what kind of coalition government might be possible.
1. The conservative/liberal coalition
At the beginning of the campaign, it looked like a formality that this would be the result.
The conservatives (Christian Democrats/Christian Social Union - or CDU/CSU) were about 20 points ahead in the opinion polls, and seemed ready to cruise into power with the liberal Free Democrats (FDP), reviving the classic centre-right coalition that Helmut Kohl ruled with for 16 years.
1. Christian Democrats/Christian Social Union (CDU/CSU): 225
2. Social Democrats (SPD): 222
3. Free Democrats (FDP): 61
4. Left Party: 54
5. Greens: 51
This is what the conservative candidate for chancellor, Angela Merkel, would have preferred, because the CDU/CSU and the FDP are close on a range of policy issues - but the two parties' combined number of seats is short of a governing majority.
On the economy, they share the same broad agenda of reforming labour market law, tax, and the welfare system.
On foreign policy, they both stand for a return to a close alliance with the US.
There are differences between the two sides. The FDP has opposed the CDU/CSU plan for increasing sales tax.
It wants to be more radical than the conservatives in reforming the economy, and also has a stronger attachment to civil liberties that could clash with the CDU/CSU's tough stance on internal security.
2. The Grand Coalition
Because the CDU/CSU cannot form a majority with the FDP, it may have to negotiate a "Grand Coalition" with Chancellor Gerhard Schroeder's Social Democrats (SPD).
On a regional level, this is not an unusual alliance. The two parties already run regional governments together in Saxony, Schleswig-Holstein, Bremen and Mecklenburg-Pomerania.
But at national level it has only happened once - in 1966-69. Historians are divided on the results, but in any case it is hard to draw meaningful parallels between now and then.
The main parties disagree over economic reforms
On policy, there are areas the two parties could agree on.
Mrs Merkel's idea of raising VAT by 2% and using the proceeds to reduce the tax burden on employers has been heavily attacked by the SPD in the campaign, but senior SPD leaders - including Mr Schroeder himself - have in the past publicly pondered the idea.
The two parties have also agreed during the life of the current parliament on a number of issues. The CDU/CSU voted for the government's Agenda 2010 labour market and welfare reforms, for instance.
They also agreed at a so-called "jobs summit" earlier this year on tax cuts.
But there is still the fear that neither side will really get what it wants.
They each have very different concepts of how to reform the health service, for instance. It is likely that here they would have to fall back on a compromise that they have already negotiated, in 2003.
The conservatives would be unlikely to get their proposals to relax protection from dismissal laws past the Social Democrats.
The same goes for their plans to further cut income tax and cut back trade union power.
For their part, the SPD might prove an unreliable partner. Mr Schroeder called this early election because he had lost the support of the party's left for further reforms - it is hard to imagine them supporting changes proposed by Mrs Merkel that would go even further.
The SPD would also have to strongly emphasise its left-wing credentials in any Grand Coalition, because the new left party Die Linke will capitalise on reforms it deems "unsocial" to score political points.
But while business leaders have warned that a Grand Coalition will be a do-nothing government, some observers say it could still move reforms forward, albeit at a slower pace than a conservative/liberal coalition.
3. The Traffic Light Coalition
This is something that has been increasingly talked of in recent days, and might enable Mr Schroeder to remain chancellor.
It would combine three parties whose colours match those of a traffic light: the SPD (red), the FDP (yellow) and the Greens.
Chancellor Schroeder dramatically revived the SPD's fortunes
It has never existed before, although the Social Democrats ruled with the FDP in 1969-82.
The thinking behind this coalition is that the FDP would be desperate enough for power after seven years in opposition that they might be persuaded to join.
But there are huge difficulties.
The FDP's radical economic reform policies would have little place with the SPD and the Greens.
At best, the party could hope to influence the debate while pushing forward some of its civil liberties agenda.
FDP ideas about relaxing rules on genetic research or introducing GM crops to Germany would also meet stiff resistance.
The FDP leader, Guido Westerwelle, has ruled out this option. On a personal level he is also known to have very poor relations with leading Greens, including Joschka Fischer.
Most speculation about this coalition includes an internal party putsch against Mr Westerwelle - but even then the policy differences make it a tough bet. It would be a fascinating coalition to watch in action!
Like the Traffic Light, this coalition would enable the current government to continue while bringing an extra member - in this case the new Left party Die Linke.
What speaks for this coalition is that all the parties describe themselves as "left". But in practice there is much that divides them.
Die Linke is a mix of reformed East German communists (the PDS) who have always opposed Chancellor Schroeder's reforms, and former SPD members who left the party because they regarded Mr Schroeder's policies as a betrayal of their principles.
Foremost among these is Oskar Lafontaine. Once the leader of the SPD, he unexpectedly resigned from his party post and from the position of finance minister in 1999.
Mr Schroeder has not spoken to him since, and rarely even refers to him by name.
Another problem with Red-Red-Green is that many people in the SPD and Greens, particularly from the former East Germany, could not stomach working with the PDS.
On the other side, Die Linke itself has said it wants to be parliamentary opposition - only aspiring to a role in a ruling coalition at a later date.
5. Other options
There are other options, although they are considered less likely and not discussed as much.
For instance the CDU/CSU could conceivably team up with the Greens, possibly also with the FDP. Some Greens and conservatives have spoken about this, but it is considered one for the next generation.
There is also the idea of Die Linke tolerating a minority SPD-Green government, something that has also been ruled out, but is perhaps more likely than a Merkel/Fischer tandem.
Politicians on all sides have been very fond of ruling out anything other than the most obvious constellations - CDU/CSU with the FDP, or Red-Green continuing.
But the results mean these will not be possible, and German politicians are being forced into some creative thinking.