What may be done for the new born on Shabbos?
The child which is born on Shabbos may have all of his needs done for him.
- Bathe him,
- Salt him in order so his flesh harden,
- Bury the placenta in order to warm the infant,
- Cut the umbilical cord,
As all these matters the infant has pain from them if they are not done for him, and since they do not contain any Biblical work the Sages allowed it [to be done].
- Straightening the child’s limbs:
One may straighten the limbs of a livable child which have dislocated due to the pain of the birth. However this is only allowed to be done on the day of the birth, however afterwards it is forbidden.
The reason for this is: because it appears like building.
To wrap him in his clothing: However to wrap him in his clothing in order to straighten his limbs is always permitted as this is the ordinary way of doing things during the week and it does not appear like building on Shabbos. However an infant which is not wrapped during the week and his limbs have bent on Shabbos, it is forbidden to wrap him in order to straighten them.
- Lifting up the foreskin of the throat:
If the foreskin of the throat of the infant fell it is permitted to place a finger into his mouth and remove the foreskin to its place even though that at times this will cause him to vomit.
If the child was born prematurely: 
Certainly born in 7th or 9th: The above allowance applies by a livable baby, which is defined as a baby which was for certain born in the ninth or seventh [month] (such as if the husband separated from her after [the original] marital relations [which got her pregnant]).
Questionably or Certainly born in 8th: However if the child was born in the eighth month, or if it is questionable whether it was born after seventh or eighth, then one does not desecrate Shabbos for him whether for the above matters, whether for matters which lack of performing can endanger him, such as to undo a mound that is over him, unless his hair and nails have completed growing, in which case he certainly was fully developed by the seventh months and it just delayed leaving his mother’s womb after having been fully developed. Therefore even if he is for certain born in the eighth, such as if her husband separated from her after having marital relations, his status is like one who was born in the seventh month for all matters and he is [thus] to be circumcised on Shabbos.
However if his hair was smitten and his nails are not complete as [are to be] when born, then even if he is questionably a child of the eighth month, his ruling is like one who is for certain of the eighth month for all matters. Furthermore even regarding life threatening matters in which one [is ordinarily allowed to] desecrate [even] when there is a doubt, even by a fetus which never had a living status due that we suspect that perhaps he has developed enough months and his hair and nails are complete, nevertheless here where we see that [his nails and hair] are not complete it is close to certain that he was born in the eighth [month], [and he] would not have been complete until the ninth month, and he hurried to be born prior to the time and he is a stillborn.
Moving the eighth month baby who did not complete his nails/hair: Even to move such a baby is forbidden the same way it is forbidden to move a stone. Therefore one does not circumcise this child on Shabbos.
The reason he is not allowed to be circumcised on Shabbos: We do not say let us circumcise the child in any event as if the child is a seventh month old then his circumcision overrules Shabbos and if he is an eighth month child then there is no Shabbos desecration here in the circumcision, as it is like merely cutting meat. [The reason for this is] as since it is close to certain that it is an eighth month child the Sages did not want to allow moving him on behalf of his circumcision in a scenario where it is close to certain that his circumcision is meaningless.
Other Opinions: There are those which argue on this and say that since it is possible that he is a seventh month child and all seventh month children are not stillborns despite that their hair and nails have not fully developed, therefore there is no prohibition at all to carry him on Shabbos, and so too he is to be circumcised on Shabbos in any event. 
According to all cannot desecrate Shabbos for this child: However one may not desecrate Shabbos for him according to all whether for matters in which there is danger [for the child] if they are not done, whether for matters that [the Sages] permitted to do for the child on the day that he was born due to his pain even though that [doing these actions] do not contain a Biblical prohibition.
The custom today-To circumcise and move babies born in eighth: Based on this [latter] opinion today we circumcise all children even if he was born in the eighth month as we are not expert in the months of pregnancy as at times a woman conceives near her immersion and at times she does not conceive until later and it is possible that the child is a seventh month child, and even though that his hair and nails have not fully developed he is to be circumcised, and so too it is permitted to move him on Shabbos.
The Final Ruling: (Regarding the final ruling in a Rabbinical matter one follows the lenient opinion and therefore one who is lenient to move the child like the latter opinion does not lose [following religious law], however with regards to circumcision on Shabbos it is better to sit and do nothing, as it is proper to suspect for the words of the forbidding opinions, [as] although there is no prohibition in moving the child there is a prohibition in circumcising the child on Shabbos for no need as similarly there is a prohibition in cutting the umbilical cord on Shabbos even according to the latter opinion as was explained [above].) [In 331/3 Admur plainly rules that an 8th month child who has not fully developed his hair and nails is not to be circumcised.]
If the baby’s hair or nails are not completely developed and it is questionable whether the baby was born in its eighth month, and certainly if one knows that it was born in the eighth month, then one may not desecrate Shabbos even for Rabbinical matters, even to help save this child. However one may be lenient to move the baby, and some are accustomed to even circumcise the baby on Shabbos, although regarding circumcision it is better to abstain from doing so on Shabbos.
If the child has complete hair and nails, or it does not but one knows for certain that it is a ninth or seventh month baby then Shabbos is to be desecrated to help save him.
Practically, may one today desecrate Shabbos to help save a premature baby if it is in the eighth month?
Yes, as today we have incubators and other forms of technology which highly raise the child’s ability of survival and it is thus no longer considered like a stillborn.
May one desecrate Shabbos to help save a test tube baby [IVF]?
While the egg is in the stage of being fertilized in the tube one may not desecrate Shabbos to help save it, as only a fetus in its mother’s womb is considered a livable entity.
 This had a supernatural effect in warming the child [M”B]
 This follows the opinion of the Michaber in Yoreh Deah 266/11
 This follows the opinion of the Rama in Yoreh Deah 266/11
 To note in 331/3 Admur writes plainly that he may not be circumcised on Shabbos.
 Piskeiy Teshuvos 330/6
 So rules Minchas Yitzchak 4/123; Kinyan Torah 3/42; Chazon Ish Yoreh Deah 155
 Piskeiy Teshuvos 330/7
 So rules Sheivet Halevy 5/47