Fred Pearce has a nice article (in Grist of all places) about how the Population Bomb essentially defused itself.
For a start, the population bomb that I remember being scared by 40 years ago as a schoolkid is being defused fast. Back then, most women round the world had five or six children. Today's women have just half as many as their mothers -- an average of 2.6. Not just in the rich world, but almost everywhere.
This is getting close to the long-term replacement level, which, allowing for girls who don't make it to adulthood, is around 2.3. Women are cutting their family sizes not because governments tell them to, but for their own good and the good of their families -- and if it helps the planet too, then so much the better....
And China. There, the communist government decides how many children couples can have. The one-child policy is brutal and repulsive. But the odd thing is that it may not make much difference any more. Chinese women round the world have gone the same way without compulsion. When Britain finally handed Hong Kong back to China in 1997, it had the lowest fertility in the world -- below one child per woman. Britain wasn't running a covert one-child policy. That was as many children as the women in Hong Kong wanted.
This is almost certainly one of those multiple-cause things, and we have always had the hypothesis that wealth and education reduced population growth.
But the author makes an interesting point, that urbanization, even in poorer countries, may a big driver as well. After all, in the city, food and living space for children are expensive, and there are fewer ways children can support the family (I hadn't thought of this before, but I wonder if industrial child-labor restrictions, which mainly affected cities, had an impact on birth rates by making urban children less lucrative?) In fact, urban jobs require educations which are expensive (even if they are free, non-productive family members must be fed and housed for years).