Scientists have recently developed a technique to screen unborn babies for over 3000 different genetic diseases using a drop of the mother's blood, without the need for an invasive amniocentesis. The test will be generally available in a few years.
There are concerns with this approach, however, that need to be more widely discussed.
First, we now know that diseases are often caused by rare genetic variants, not common variants. While certain genetic variants are well-known to cause disease, with other variants it's less clear. Maybe a variant will causes minor problems, or maybe none at all, so how do you interpret the screening results?
Second, the screening will be conducted after a fetus is conceived and developing. Parents will often want to abort if a genetic disease is detected, even if a rare variant with a low chance of severe disease is found. The number of abortions will thus greatly increase.
It's a morally grey area to abort before the viability of the fetus (healthy births have been recorded for pregnancies shorter than the legal abortion term). Abortions are also painful, costly and dangerous. I believe this will inevitably lead to greater demands for in vitro fertilization in the future, when screening can be done against multiple fertilized eggs before they are implanted, instead of aborting after pregnancy begins.