Indian mythology is full of wondrous birth incidents. Science cannot explain such births, one just has to believe on them. All of them cannot be listed here together, so here are listed only some such births.
There was a king in olden days, named Kritveerya. he was very rich and a good king. When he died his children became king. Once they needed some money. They heard that Bhrigu Vanshee Braahman had lots of money so they went to them to ask for some money as beggars. Seeing this some of them gave some money to them, some hid their money and left their houses, some just ran away. Kshatriya knew this so they started searching their houses. One Kshatriya found a lot of money while digging in a Braahman's house, seeing this all Kshatriya got very angry and they started killing them. Their women started running away from there. One of them was with the child in her thigh. A Braahman woman complained about her running away so they started following her.
While she was running away, and the Kshatriya were following her, the child came out of her thigh. He was shining like a Sun. His shine was so strong that it made all his followers blind. Now they could not see anything, but they understood that it happened because of that woman, so they went to that woman and asked for her forgiveness. They also told her that if they would get their eyesight back they would go back and would never commit such sinful acts. She said - "I am not angry with you or anybody. It is only this child who is angry with you on killing his forefathers. If you please him he might give your eyesight back. I have protected him in my thigh for more than 100 years. He knows all Ved." So they prayed him, he got pleased and gave their eyesight back. After getting their eyesight back, they went away. Because of being born by tearing the thigh, he was known as Aurv.
Maharshi Jamadagni and Vishwaamitra
This story shows why Vishwaamitra Jee became a Brahmarshi even being a Kshatriya and why Parashuraam Jee became a Kshatriya even being a son of so Gyaanee Braahman. There was a great Rishi named Richeek. His wife's name was Satyavatee who was the daughter of King Gaadhi of Kushik Vansh. After a while both Satyavatee and his mother requested Rishi Richeek to give them a son, so the Rishi prepared two Charu (Kheer) - one for Satyavatee and another for her mother and told this to his wife. Satyavatee's mother thought that Rishi must have prepared better Charu for his wife, so she asked her daughter to give her her own Charu and she gave her own Charu to her daughter.
When Richeek Muni knew this, he said to his wife - "I prepared both Charu separately keeping in mind that since you are a Braahman's wife so you should have a Brahm Gyaanee son and your mother is the wife of a king so she should have a son possessing good Kshatriya qualities. Now since you have exchanged your Charu, now you will have a Kshatriya son and she will have a Brahm Gyaanee son." Hearing this Satyavatee started crying, asked for his forgiveness and said - "I don't need a Kshatriya son, have mercy on me." Richeek Muni said - "All right, Now not your son will be like this but your grandson will be like this."
So Satyavatee gave birth to Maharshi Jamadagni and her mother gave birth to Vishwaamitra Jee who even being born in a Kshatriya family attained the status of Brahmarshi. Jamadagni's son was Parashuraam who in spite of of being born in a Braahman family emptied Prithvi from Kshatriya 21 times and created five tanks filled from their blood which came to be known as Samant Panchak.
Saaraswat Rishi was the son of Maharshi Dadheechi. People know him in the reference that his bones were so strong that they were even stronger than any weapon, that is why Indra was advised to ask for his bones to make his Vajra (thunderbolt) to kill Vritraasur Raakshas, because he was not killed by any other weapon. How his bones were so strong, it is because of his Tap (penance).
Once he was performing Tap that Indra, as usual, got worried that Dadheechi would get his kingdom, so he sent Alambooshaa Apsaraa to distract him from his Tap. She got successful in her objective. Seeing her Muni's semen fell in Saraswatee River. As it fell, Saraswatee River kept it in her womb. When the time came she gave birth to a baby boy. She brought it to Rishi and said - "This is your son, please accept him." Dadheechi got very happy to see him. He gave him to her and named him Saaraswat on his mother's name. Later once a severe famine occurred in the country so he then fed the people with the help of Saraswatee.
Maharshi Vashishth, Maharshi Agastya and Dronaachaarya
Originally Vashishth Jee was the brain child of Brahmaa Jee. Once he got angry with the King Nimi, both cursed each other and both left their mortal bodies. Later he was reborn from Mitraavarun and Urvashee Apsaraa.
Once Mitra and Varun were going somewhere that they saw Urvashee Apsaraa. Seeing her their semen fell. They immediately kept it in a pot. And from that pot two Rishi were born - first Vashishth Jee and later Agastya Jee. Both are thus called Kumbhaj - born from a Kumbh (pitcher).
Dronaachaarya was the son of Bharadwaaj Rishi. Drone means a pot. He was named Drone because he was born not from human body but from outside the body, from a pot. Once Bharadwaaj Jee was going to Gangaa River that he saw Ghritaachee Apsaraa who came there to take bath. Seeing her the sage was overcome with a desire to have her so his semen fell, but he collected it in a vessel (called Drone also), so from that fluid was born his son, and since he was born from a vessel, he came to be known as Drone.
There was a great Muni Uddaalak who had two sons - Nachiketaa and Shwetketu. he had a daughter also, named Sujaataa, whom he married to Kahod (or Kahol) - one of his very good disciples. Kahod also used to teach his disciples, so Sujaataa used to sit in their classes for the good of her unborn baby. Once Sujaataa was sitting in Kahod's class and Kahod was reciting some verse that the child from the womb spoke - "This is not the way to recite this verse, O Father." Hearing this Kahod got very angry but remained silent. When it happened 8 times, he could not tolerate his insult anymore and cursed him - "You will be born deformed from eight places of your body." So when the child was born he was crooked from eight places of his body - two feet, two knees, two arms, chest and head, and thus was named as Ashtaavakra.
Satyavatee was the wife of Raajaa Shaantanu. There was a king named Uparichar Vasu. He loved his wife very much. Once she expressed her desire to the king to have a child, but at the same time His Pitri came and asked for some food. So he went to forest to bring some good meat for them. But he remembered his wife, so he gave his seed to an eagle to carry it to his wife. The eagle took it in his beak and flew away. Another eagle saw it carrying something which it mistook as a meat piece so it tried to snatch it from its beak. In this process the semen fell down. Yamunaa river was flowing down, so it fell in the river. Immediately a fish swallowed it. The fish was an Apsaraa named Adrikaa.
A fisherman caught that fish but as he came to know that that fish was carrying he did not kill her and took great care of her. When the time came she gave birth to a twins - one girl and one boy. The fisherman took them to the King and the King recognized them as his own children. He gave the daughter to the fisherman and kept the boy with him. Later he appointed the boy as his Army General. The girl was thus brought up by the fisherman and was named as Satyavatee, but being living in fisherman's community she smelled like a fish so she was known as Matsyagandhaa.
How Kaurav were born? Shaantanu had two sons from Satyavatee - Chitraangad and Vichitraveerya. Unfortunately Chitraangad was killed in a battle with a Gandharv of the same name on a trivial matter even before he was married. Vichitraveerya was married to Ambikaa and Ambaalikee but he also died shortly after his marriage without having any child. Then Satyavatee asked her eldest son Ved Vyaas Jee to produce children from Vichitraveerya's widows. he produced two chldren from them - blind Dhritraashtra and Paandu. Dhritraashtra was married to Gaandhaaree and Paandu was married to Kuntee and Maadree. Dhritraashtra's son were called Kaurav and Paandu's son were called Paandav. Everybody knows that Paandav were the sons of Devtaa, but how Kaurav were born is interesting to know.
Gaandhaaree conceived before Kuntee conceived her first child, and thus was expecting her children before Kuntee's child. But she bore her first pregnancy for 2 years and no child was born to her. She was tired of this. In the meantime she heard that Kunte had given birth to a son, so she got more perturbed. Grieved with her situation, she struck her stomach with a stone and a mass of flesh as hard as an iron ball came out of her womb. As she was about to throw away, Vyaas Jee came there and asked her - "What did you do this?" Gaandhaaree said - "Hearing that Kuntee has a son I filled with grief and I struck my womb with a stone and this ball came out. You said that I would give birth to 100 sons, but this is only a mass of flesh?"
Vyaas Jee said - "You did not believe me that is why you did this. Now you get 100 pots of Ghee immediately and keep them in shady place. In the meantime pour cool water on this iron ball." Gaandhaaree got 100 pots filled of Ghee and poured cool water over the ball. As the cool water was poured on the ball it split into 101 pieces, each of the size of a thumb. Vyaas Jee kept one piece each in 100 pots and asked Gaandhaaree to open the pots after full 2 years. Gandhaaree's 100 sons and 1 daughter were out of those pots. According to their order of birth, Duryodhan was their eldest son.
Apart from these wondrous births, there are other births also - Raajaa Prithu, Jaraasandh, Shishupaal, Makaradhwaj etc. I will write about them sometime later.