Pope John Paul II was renowned for his social conservatism. Some Catholics say it is time for the Church to modernise. Others say the new Pope should continue where the last left off.
Author Dr Lavinia Byrne, a former nun and supporter of the ordination of women, puts forward her point of view, followed by Italy's Europe Minister, Rocco Buttiglione, a friend of the late Pope John Paul II, who failed to become an EU commissioner in 2004 after describing homosexuality as a sin.
Pope John Paul II was a genuinely charismatic figure. On an ocean liner, he would have been playing with the band, entrancing the passengers or faithful, waving to passing ships from other Churches and faith communities.
The next Pope will have to go down into the engine room and grapple with murky problems about the balance of power between the papacy and the Vatican, and between the power of the centre and of the edges - namely the local or national Churches and their bishops.
The Church is too centralised. Although John Paul II helped to defeat communism, the Church he left behind is far more totalitarian than the Church he inherited. He allowed the Vatican to become, in its own way, a Kremlin.
Only when this unglamorous work of re-organisation in the engine room is done will the Church be able to look intelligently at the problems that beset it.
In order to tackle the problem of secularisation in Western Europe, the Church must be open to change.
It needs a new discourse about human relationships to respond to problems such as the dissonance between the Church's teaching on contraception and the practice of most of the faithful, clerical celibacy, child abuse, and the spread of HIV/Aids.
Above all, it needs a fresh understanding of the place of women in the Church. The ordination of women is essential.
If the new Pope is a man of faith, a man of great prayer, the question of whether he is a bit more conservative or a bit more modern is not really so important. He might decentralise, but we don't understand what we are looking for if we think this is the main criterion.
When people say the Church should move with the times, where are the times going? I have a slight suspicion that the Church does not know, and that the times themselves do not know where they are going. They will go where responsible human beings lead them.
We should value what was great in John Paul II. He was always aware of the fact that he should be a witness and that through him the course of history could change.
I remember Poland in the 1970s; there were a lot of people who thought the Church had to follow the course of the times, to find and adjust to communism, because the times were moving towards communism. It was a cheap philosophy.
I am not an expert on the issue of the ordination of women, but nobody has the right to become a priest. It is a gift of God. I do not like the way this issue is sometimes debated. Can someone's rights be violated because they have not become a priest?
As regards homosexuals, they may be ordinary, natural people, but ordinary, natural people are sinners. We all stand in need of God's mercy.
Many Catholics, since the 1st Century after Christ, have been sinners, but the Church was not in agreement with this. It is not an argument to change the doctrine of the Church. To say that it is seems very naive.
Dr Lavinia Byrne is author of Woman at the Altar, published by Hodder and Stoughton. Rocco Buttiglione is the author of Karol Wojtyla: The Thought of the Man Who Became Pope John Paul II, published in English by Wm B Eerdmans Publishing Company.