They looked forward to the triumph of "aggiornamento" – a bringing up to date, a coming to terms with the modern age, for the church. Others were less pleased.
Archbishop Giovanni Montini feared it would stir up a "hornets' nest".
At the time, he was considered a moderniser.
But when he became Pope Paul VI, he began what looked like a counter-revolution.
Since then, the Holy Roman Church has grown ever more authoritarian and intolerant, more insistent on obedience to doctrine and more narrow in its style of governance.
This last week, everything changed with Pope Francis's sensational interview with the Jesuit newspaper 'La Civilta Cattolica'.
His stand is far more radical than anything that emerged from Vatican Two. He says not all doctrines are of equal importance. He objects to the obsession with abortion, gay marriage and contraception.
And he says the church is not a "small chapel" for the select few. Unless it changes, "the moral edifice of the church is likely to fall like a house of cards".
Statements like these are not merely amazing. They are revolutionary.
They can reasonably compare with the turmoil of religious – and political – thought that gave the world Martin Luther and the Protestant Reformation in the 16th Century.
But the world we now live in is a different world from that of Luther.
We can see Francis and hear him speak. We may view him with admiration or with the same scepticism and apathy with which we react to politicians. In either case, we think we know him.
What do we think we know?
We know Francis loves God. He loves humanity. He loves gays and atheists. He loves the poor. He has pondered the mysteries of the universe. When he warns that the moral edifice of his own church could collapse, we listen.
But we do not know if he can save it from its present misgovernment. His job would be awesome in the easiest of times. At a time like the present, the task he has set himself borders on the superhuman.
He has enemies.
They will fight back, fortified by the knowledge that millions like the comfort provided by rigid doctrine and authoritarian rule.
As a Latin American, he must often have looked northwards with something like dismay.
In the US, the church is frequently allied with the most backward elements: fundamentalists, creationists, climate-change deniers, the right-wing now dominant in the Republican party. These have powerful friends, and access to limitless funds.
Closer to his new home, Francis faces indifference or hostility in countries shocked and baffled by economic disaster.
He has begun to reform the Curia and restore "collegiality".
But he cannot solve questions of doctrine by inviting Catholics and others to stop talking or thinking about them.
He has opposed the ordination of women. That question will not go away. Neither will that of married clergy.
In Europe and North America,
birth control ceased to be an issue long ago.
Not so in the Third World.
In South America, easily available contraception has done marvels for the health and status of women.
But in Africa, many millions of women die for lack of it.
Not so in the Third World.
In South America, easily available contraception has done marvels for the health and status of women.
But in Africa, many millions of women die for lack of it.
And
in Ireland?
The Government recently published its updated policy on Third World aid. The paper devotes half a sentence to contraception.
What influence prompts a (supposedly) developed country to run away from a question of such importance?
The Government recently published its updated policy on Third World aid. The paper devotes half a sentence to contraception.
What influence prompts a (supposedly) developed country to run away from a question of such importance?
Still, in the reign of Francis there is hope, even for Ireland.
Archbishop Diarmuid Martin professes himself "very encouraged".
Fr Brendan Hoban of the Association of Catholic Priests is "absolutely exhilarated".
Let
us see now if Fr Hoban's colleagues who have been silenced by the
Vatican have their rights restored.
And let us see if the Pope's wise words attract an audience willing not only to hear them, but to act on them.
If that happens, it will effect fundamental changes for the better, not only in the church, but throughout society; and Francis will be seen, rightly, as a prophet.
He will have brought about something so long-desired and so long-delayed – the Catholic Reformation.
And let us see if the Pope's wise words attract an audience willing not only to hear them, but to act on them.
If that happens, it will effect fundamental changes for the better, not only in the church, but throughout society; and Francis will be seen, rightly, as a prophet.
He will have brought about something so long-desired and so long-delayed – the Catholic Reformation.