One of the cartoons shows the Prophet wearing a headdress shaped like a bomb. In another he says paradise is running short of virgins for suicide bombers. Islamic tradition bans depictions of the Prophet or Allah.
Eight commentators give their views on the controversy to the BBC.
Click on the links below to read what they have to say.
Dr Yunes Teinaz is the spokesman for the London Mosque and Islamic Cultural Centre
The European newspapers have published extremely offensive caricatures of the Prophet Muhammad. They are humiliating and racist. Muslims love the Prophet more than anyone - even their own families - and have a very strong belief that he is the messenger of God.
We respect the heroes of other religions and we would expect the same from the followers of other religions and ideologies. No Muslim, for example, is allowed to portray a picture of Jesus.
We do value the freedom of expression in Europe, but it shouldn't be abused to provoke hatred and division between communities. Freedom of expression is not a licence to attack a culture or religion.
Muslim governments have the right to boycott Danish and Norwegian goods. On the other hand, the way forward is to solve this amicably. Any violent threats against Danish, or other, people are completely unacceptable.
We believe the governments that allowed these freedoms to be abused should apologise to the Muslim communities.
Flemming Rose is the culture editor of Jyllands-Posten newspaper in Denmark, which originally published the cartoons
I did not ask the illustrators to make the Prophet a laughing stock - I asked them to draw the Prophet as they see him.
In Denmark we have a tradition of satire and humour and some cartoonists made satirical cartoons. We have done the same thing with Jesus Christ and other religions. That's what we do with the royal family, politicians and other public figures. We were not treating Islam or the Prophet any differently from how we treat everybody else in Denmark.
The cartoons have given impetus to a very important debate about integration in Denmark. The debate on the one hand looks at how much the receiving community should compromise on their own values and standards when they are receiving foreigners, immigrants and refugees.
On the other hand the debate focuses on how much of their own culture immigrants have to give up.
Stewart Lee is the creator of Jerry Springer - The Opera, which some Christians found offensive
When you satirise something, you need to at least give it the credit of understanding it. The cartoon in which the Prophet is trying to stop suicide bombers entering the afterlife because they have run out of virgins has a kind of political point behind it. But I don't think they really appreciated the massive taboo you cross by portraying the image of Muhammad. There is really no historical precedent for it.
They have tried to deal with a subject they don't know enough about, and this is one of the teething problems of the cross-over of cultures in the world. I'm sure the level of offence is far greater than would have been intended.
I look forward to a point where we can live in a genuinely multicultural society when people know enough about each other's faiths and culture to be able to satirise these things from an informed position.
In Jerry Springer - The Opera, we were looking for a story that could be commonly understood in a Christian context. In the West, Christianity relinquished the right to be protective of its icons the day Virgin Mary snow globes were put up for sale at the Vatican.
But in the Islamic culture it is very different. To use a corporate image, Islam has always been a lot more conscientious about protecting its brand and as a satirist you need to engage with it on its own terms. That's what we did with Jerry Springer with the Christian religion.
Munira Mirza is a commentator on multicultural issues and Islamophobia
British newspapers should publish the images. Muslims should be able to see them and judge them for themselves, that's why we have freedom of speech.
Many Muslims want the same freedoms as everyone else to debate, criticise and challenge their religion.
They want to be able to say: "Hey we're not children, we can handle criticism, we don't need special protection - we're equal."
Many don't want to be treated as a special group, seen as worthy of more protection from criticism than other groups because of their apparent victim status.
There are a lot of British Muslims who I'm sure would not be offended by the cartoons. There are, of course, many who are upset and hurt, but that's the point of living in a free society.
No matter the price, the principle of freedom must be defended. Unless we stand up for freedom of speech, we are unable to engage freely and hold belief systems - of all kinds - to account.
In Denmark, there are counter-demonstrations by moderate Muslims saying they don't want the images banned.
This idea that all Muslims have to hold the line against Islamophobia is just nonsense. We should not play the games of extremists and nor should we play into the very patronising assumptions of the British political elite about what Muslims are capable of listening to.
Sayeed Nadeem Kazmi is a member of the Al-Khoei Foundation, an Islamic charitable organisation based in London
The cartoons do seem to be insulting to me and, although I don't know the motives behind them, they could have been avoided.
I understand satire and I don't think satire has any limits but these are verging on the racist.
They have a very 19th Century depiction - the attire conjures up the demon Muslim - but they are given a very 21st Century twist - that Islam is a big threat.
The illustrator may have been testing the waters but I think that was dangerous and irresponsible in the current climate.
I embrace freedom of expression but with freedom comes responsibility. We embrace debate and there are many questions that can be asked about Islam that are very legitimate, such as the question about whether it is violent or not. But there are legitimate ways to conduct a debate - and this was not one of them.
We have to be very aware in a multicultural, multi-faith and multi-ethnic society. It is a time of great crisis between communities.
Muslims are feeling under siege since 9/11 and the London bombings last year.
It is interesting that these cartoons were first published in Denmark, where there is an official Nazi party.
Karen Armstrong is a leading British commentator on religious affairs and author of Muhammad: A Biography of the Prophet
Each side needs to appreciate the other's point of view. I think it was criminally irresponsible to publish these cartoons. They have been an absolute gift to the extremists - it shows that the West is incurably Islamophobic. It sends a very bad message.
But, more seriously, it is letting ourselves down. We trumpet abroad about what a compassionate culture we are. But these cartoons depicting Muhammad as a terrorist are utterly inaccurate, feeding into an Islamophobia that has been a noxious element in Western culture since the time of the Crusades. It can only inflame matters at this very crucial juncture of our mutual history.
On the other hand, in a secular Europe, freedom of speech has developed as one of our sacred values. We fought hard for it, but we have to remember it carries responsibilities. For example, do we have a right to say whatever we want even if it is false and dangerous?
More importantly, however, freedom is as sacred a value for us as the Prophet is to Muslims.
We are seeing here a clash of two different notions of what is sacred and this is part of the modernising process.
Modernisation and secularisation has this bumpy ride where people at different levels of modernisation are clashing. In other parts of the world where modernisation is not yet complete it is not regarded as a crucial as other sacred realities.
And now we are all living in this multicultural society cheek-by-jowl with one another, not even within a single country but we are linked to one another in our global village. We have to learn to live side by side better than this.
Roger Koppel is the editor of the German newspaper Die Welt, which reprinted some of the cartoons
It's our cultural tradition. I think it is legitimate to publish pictures and cartoons like this; I don't think they go too far. Of course, my personal opinion is that the reactions have gone way, way too far.
We did it because it's a highly political event of course and we wanted to show what is the corpus delicti, the reason for all this uproar - that was the first thing. And the second thought was that we think that we are living in a secular society where even religion can be subjected to criticism and satire.
It's not acceptable in a western country, if you publish a cartoon like this, that the newspaper has to apologise, or even the prime minister has to apologise.
Also, Muslim spiritual leaders seem not to be doing much in order to calm the reaction of their people, in order to explain, for example, what it means in our tradition to publish cartoons like this.
Dr Lam Akol is the Sudanese foreign minister. Sudan called for a boycott of Danish goods and called off the visit of a Danish minister
There is a problem in that you need to be mindful of what other people think and believe and feel. The world is becoming like a village, you can't suppress what is being published in the West from the Third World.
It is very important when people want to promote religious dialogue and co-existence to avoid such issues.
In the Third World they hardly separate between what is a journalist and what is the Danish government's point of view.
Once a Danish paper has published something then it is concluded that this is the opinion of everybody in Denmark. So that is the kind of feeling that should have been understood from the beginning.
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The Iranian newspaper  Hamshari daily has stated that it will publish anti-Semitic cartoons in response to the Danish Mohammad cartoons. The newspaper, owned by the Tehran city council, says that the anti-Semitic cartoons will lampoon the Holocaust, following denials by the Iranian government that the Holocaust even happened, and will show "America and Israel's crimes and plundering". The newspaper has launched an international competition to find the most suitable caricatures about the Holocaust.

Farid Mortazavi, the paper's graphics editor, said to the Guardian newspaper that "''The western papers printed these sacrilegious cartoons on the pretext of freedom of expression, so let's see if they mean what they say and also print these Holocaust cartoons''." 

Mr Mortazavi said that the February 8 edition of the paper would invite cartoonists to enter the competition, with gold coins as prizes for the 12 winning artists -- the same number of cartoons that appeared in the conservative Danish newspaper Jyllands-Posten lighting the touchpaper for fury which has swept around the Islamic world.
According to the article alongside the original "Mohammad cartoons" in the Jyllands-Posten newspaper in late September 2005, the cartoons were intended to show that:

:''The modern, secular society is rejected by some Muslims. They demand a special position, insisting on special consideration of their own religious feelings. It is incompatible with contemporary democracy and freedom of speech, where you must be ready to put up with insults, mockery and ridicule. It is certainly not always equally attractive and nice to look at, and it does not mean that religious feelings should be made fun of at any price, but that is less important in this context. ... we are on our way to a slippery slope where no-one can tell how the self-censorship will end. That is why Morgenavisen Jyllands-Posten has invited members of the Danish editorial cartoonists union to draw Muhammad as they see him. ...'' 

Iranian president Mahmoud Ahmadinejad has criticised the argument of freedom of speech employed by European newspapers to justify publication of the cartoons.

:''If your newspapers are free why do not they publish anything about the innocence of the Palestinians and protest against the crimes committed by the Zionists?'' the Mehr news agency quoted him as saying.

Anti-Semitic cartoons will be chosen rather than anti-secular or anti-Christian cartoons for the response to the Danish newspaper despite the fact that Denmark has only a  small Jewish population of  between 6000 and 7000.

In related events, Mr Jahjah of the  Arab European League have launched a similar ''Freedom of Speech Campaign'' in which they hope to demonstrate that European News Media are only willing to publish anti-Muslim cartoons, and will not publish anti-Christan or anti-Semitic cartoons. The first of these cartoons (drawn by Nabucho) is shown on this page. 

Paul Belien of the Brussels Journal noted that "The three AEL cartoons posted so far have been very instructive in that they have all mocked the Nazi persecution of the Jews and the Holocaust, as if Mr Jahjah wants to emphasize that islamofascism is indeed the ideology he adheres to."