Blood groups are typically not a problem between parents.  The baby's blood will be O, A, B, or AB depending on the dominant gene.

However, rH factor is a problem.  If the mother is rH negative, and the baby is rH positive, then the mother's blood will reognize the baby's blood as an invader. This is typically not a problem for the first baby of the union, but if the baby's blood and mother's blood intermingle during delivery, then the second and previous babies may suffer as a result of rH incompatibility.

There is a shot that can be administered to an rH negative mother with a few hours after delivery that will prevent the mother from producing rH antibodies against the baby's blood.