The mythological accounts of the founding of Rome are:
Aeneas as Founder of Rome:
The Trojan prince Aeneas is sometimes credited with the founding of Rome as the culmination of his post-Trojan War adventures, but the version of the Roman foundation myth that is most familiar is that of Romulus, the first king of Rome.
Birth of Romulus and Remus
Romulus and Remus were twin brothers, the sons of a virgin named Rhea Silvia ( also called Ilia) and the god Mars, according to legend. The grandfather, Numitor, and the great-uncle, Amulius, who between them divided the wealth and kingdom of Alba Longa, a city founded by Aeneas’ son Ascanius, but then Amulius seized Numitor’s share and became sole ruler. To prevent retaliation by offspring of his brother, Amulius made his niece a vestal virgin. When Rhea became pregnant, her life was spared because of the special pleading of Amulius’ daughter Antho. Although she kept her life, Rhea was imprisoned.
Exposure of the Infants
Contrary to plan, the virgin Rhea was impregnated by the god Mars. When the twin boys were born, Amulius wished to have them killed, and so bid someone, perhaps Faustulus, a swineherd, expose the boys. Faustulus left the twins on the river bank where a she-wolf nursed them, and a woodpecker fed and guarded them until Faustulus took them into his care again. The two boys were well educated by Faustulus and his wife, Acca Larentia. They grew up to be strong and attractive.
"They say that his name was Faustulus; and that they were carried by him to his homestead and given to his wife Larentia to be brought up. Some are of the opinion that Larentia was called Lupa among the shepherds from her being a common prostitute, and hence an opening was afforded for the marvelous story.
Romulus and Remus Learn Their Identity
As adults, Remus found himself imprisoned, and in the presence of Numitor, who determined from his age that Remus and his twin brother could be his grandsons. Learning of Remus' predicament, Faustulus told Romulus the truth of his birth, and sent him off to rescue his brother.
The Twins Restore the Rightful King
Amulius was despised, and so Romulus drew a crowd of supporters as he approached Alba Longa to kill the king. The twins re-installed their grandfather Numitor on the throne and freed their mother who had been imprisoned for her crime.
The Establishment of Rome:
Since Numitor now ruled Alba Longa, the boys needed their own kingdom and settled on the area in which they had been raised, but the two young men couldn't decide on the exact site and started building separate sets of walls around different hills: Romulus, around the Palatine; Remus, around the Aventine. There they took auguries to see which area the gods favored. On the basis of conflicting omens, each twin claimed his was the site of city. An angry Remus jumped over Romulus' wall and Romulus killed him.
Rome was therefore named after Romulus.
"A more common account is that Remus, in derision of his brother, leaped over the newly-erected walls, and was thereupon slain by Romulus in a fit of passion, who, mocking him, added words to this effect:" So perish every one hereafter, who shall leap over my walls." Thus Romulus obtained possession of supreme power for himself alone. The city, when built, was called after the name of its founder."
Aeneas and Alba Longa:
Aeneas, son of the goddess Venus and the mortal Anchises, left the burning city of Troy at the end of the Trojan War, with his son Ascanius. After many adventures, which the Roman poet Vergil or Virgil describes in the Aeneid, Aeneas and his son arrived at the city of Laurentum on the west coast of Italy. Aeneas married Lavinia, the daughter of a local king, Latinus, and founded the town of Lavinium in honor of his wife. Ascanius, son of Aeneas, decided to build a new city, which he named Alba Longa, under the Alban Mountain.
Alba Longa was the hometown of Romulus and Remus, who were separated from Aeneas by about a dozen generations
"Aeneas was hospitably entertained at the house of Latinus; there Latinus, in the presence of his household gods, cemented the public league by a family one, by giving Aeneas his daughter in marriage. This event fully confirmed the Trojans in the hope of at length terminating their wanderings by a lasting and permanent settlement. They built a town, which Aeneas called Lavinium after the name of his wife. Shortly afterward also, a son was the issue of the recently concluded marriage, to whom his parents gave the name of Ascanius."
Plutarch on Possible Founders of Rome:
"... Roma, from whom this city was so called, was daughter of Italus and Leucaria; or, by another account, of Telephus, Hercules's son, and that she was married to Aeneas, or ... to Ascanius, Aeneas's son. Some tell us that Romanus, the son of Ulysses and Circe, built it; some, Romus the son of Emathion, Diomede having sent him from Troy; and others, Romus, king of the Latins, after driving out the Tyrrhenians, who had come from Thessaly into Lydia, and from thence into Italy."
Isidore of Seville on Evander and the Founding of Rome:
There is a line (313) in the 8th book of the Aeneid that suggests Evander of Arcadia founded Rome. Isidore of Seville reports this as one of the stories told about the founding of Rome.
"A banish'd band,
Driv'n with Evander from th' Arcadian land,
Have planted here, and plac'd on high their walls;
Their town the founder Pallanteum calls,
Derived from Pallas, his great-grandsire's name:
But the fierce Latians old possession claim,
With war infesting the new colony.
These make thy friends, and on their aid rely."