In His Name The Most High
Peace upon Muhammad & his household

1978 witnessed the birth of "test-tube" baby. Another hallmark of human advancement in science. It was a welcome achievement from every quarter except the Vatican.

ISLAM AT THE MIDDLE
There are two extreme views:
Roman Catholic church out rightly condemned In Vitro fertilization, artificial insemination and surrogate motherhood.

Libertarians adopted all these method whether done within legally married couples or otherwise.

Islam allows these techniques so long its done between married couples.

TECHNIQUES
A. Artificial Insemination By Husband (AIH)
This is when Husband can produce sperm but the sperm is inactive to achieve fertilization.
Sperm is usually collected via masturbation, improved upon in the Lab and then introduced into the wife's cervix by injection.
Islamic Shari'ah allow this so long its between two legally married couple. Masturbation is forbidden in Islam but by one's spouse, its allowed.

B. Artificial Insemination By Donor (AID)
This is similar to AIH except that women who wished to have child yet remain unmarried usually sought this method to become pregnant while any men can donate.
This is definitely not allow in Islam since it does not involve any sexual relations between the donor and the woman, and therefore cannot be categorized as adultery. The reason why its being tagged 'Haram' is the warning of the holy Qur'an:

"Say to the believing women that they...should guard their private parts"~ (24 vs 30).

The command to guard the private part is unqualified. Meaning not only from illicit sexual intercourse but from any other means except from their spouses. Qur'an says:

"Those who protect their private parts except from their spouses...therefore whosoever seeks more beyond that, then they are the transgressors."~(23 vs 5-7).

In this case, Injecting the sperm of another man is a transgression and violation of Allah's command.
Child conceive through this mean is not an illegitimate child since he was not conceive through illicit sexual relation (adultery). Child-mother relationship is clear; and would be considered the child of the donor, and will inherit him.

A question was put to Imam Jafar sadiq (a.s) that if a woman had intercourse with her husband, and she immediately pertook in lesbian relation with her slave girl. And if accidentally sperm of the husband flow from the woman into the vaginal of the slave girl and she got pregnant. What happen in this case? Imam, after explaining the punishment upon the two women, say the child born from the slave-girl will be related to, and given in custody of, the owner of the semen.

C. In Vitro Fertilization (IVF)
A situation where the woman's fallopian tubes are blocked and cannot be surgically remedied, IVF is used. Even for those who can't conceive despite having normal fallopian tubes.
In vitro means a 'test-tube.' In this case, one or more ova are removed from the mother via a particular syringe and placed in a certain test-tube. Then sperm of the husband is used to fertilized the ovum. After fertilization, the ovum is allowed to develop; it is then implanted into the mother's womb.

D. Surrogate Motherhood
It is a by-product of artificial insemination. Its a situation where a fertilized ovum of another couple is injected into another woman; and carries the child to its full term for the couple. This is done free or charged.

Although it is not allow in Islam based on the explanation above with those specific Quranic verses. But there are certain cases that might allow the permisibility.

If a woman cannot carry her husband's child, the husband can marry another woman and the fertilized ovum of the first wife can be injected into the womb of the second wife with her approval.

WHOSE CHILD IT IS?
Genetic mother (first wife) or biological mother (second wife)? Imam khomeini's verdict was that it depends on the age of the foetus. If it was inseminated in the womb of the second wife after four months, then its belong to the first wife. The second wife is just a receptacle. If it is before four months, its difficult to say it belong to the first wife (Tahrir wasila vol. 2 p. 623)
Some Marja'i ruled based on 'urf' that the woman who 'gives birth' is the mother irrespective of the age of the foetus.

Allah knows best.

References:
1. Marriage and Morals in Islam by Sayyid Muhammad Rizvi
2. Al-Fiqh alal-madhahibi'l-khamsah, by sheik Muhammad Jawad al-Mughniyyah.
3. Tahrir wasila of Imam khomeini.

Salam