At 42 weeks, most of what's happening in your womb is simply a continuation of what was happening last week, and the week before that: Lung and brain development, shedding of vernix and lanugo, and just general chilling out in about the most comfortable home ever.
Still, other goings-on might be a little bit troublesome, which is why your doctor or midwife might be raising the subject of inducing labor this week. At 42 weeks, the concern is mostly the baby's continued weight gain, because it could mean your baby grows too large for an "easy" vaginal delivery. If he or she grows big enough, your doctor might (or might not!) end up recommending a C-section.
Also this week, your baby's hair and nails are still growing, so you may be greeting a newborn who could really use a cut and a manicure.
And all the while, you're sitting around, probably brainstorming ways to start your labor. Here, according to old wives' tale, your partner might actually be able to help …