Roman religion and, therefore, Roman mythology, was a long, drawn-out process of bringing together the gods, stories, and rituals of various cultures and making them Roman. People who study religion or mythology for a living call this syncretism.
A Roman might nod to his Lares and Penates (original Roman gods) in the morning, pray to Isis (an Egyptian goddess) at lunchtime, attend a feast in honor of Heracles (a Greek hero), in the afternoon, have his future told by a haruspex (a priest who had learned the Etruscan art of liver-based fortune telling) later in the day, and ended the day attending a meeting of the cult of Mithras (a Persian deity whose followers met in caves).
By doing all of this, this Roman wasn't betraying his culture; he was being a good Roman! After all, what was the point of conquering the world if you don't have cool new gods and cool new stories to show for it?
Why the Greek pantheon?
It's no secret that the big-time Roman gods were more or less the same as Greek gods, just with different names. How did Greek gods come to Rome? They followed the Romans home, and the Romans decided to keep them!
The story goes like this: Livius Andronicus was a Greek who lived from approximately 284 to 207 B.C. He lived in the city of Tarentum, which was a city settled by Greeks in southern Italy. In 272 B.C., the Romans conquered Tarentum, and Livius became a slave to a Roman. This Roman brought his new slave to Rome, and sometime later set him free.
Livius Andronicus's name was probably, originally, Andronikos, a Greek name. When he was freed from slavery, or manumitted, he took the name of his former master, who was one of the Livii, a very old Roman family. So, he became Livius Andronicus.
After he was freed, he set himself up in business as a schoolteacher. But he also wrote literature. His biggest work was a translation of Homer's Odyssey into Latin, a work called the Odyssia. He had to work hard to bring that Greek epic into Latin. The existing Roman gods didn't always fit neatly with the Greek ones. So, he made some substitutions. For example, the Greek Muse (daughter of Zeus who inspires poets) he replaced with the Roman Camena (a goddess of a fountain). He replaced Moira (the Greek goddess Fate) with Morta (the Latin goddess Death), and so on.
By making changes like that and lots of other changes, Livius Andronicus made his Odyssia more than a cheap copy of the Odyssey. He made it into a real Latin epic poem just for Romans. And by populating the story of Odysseus with Roman gods — Jupiter, Venus, Juno, and the others — he tied those gods together closely with the Greek gods. The idea caught on, and Roman mythology suddenly became much more "Greek" than it had been before.
Imported deities from other shores
Rome got some of its gods from other sources. A big empire has lots of influences in it. Governors in the provinces and the armies that accompanied them got to experience different cultures and, often, foreign wives; and plenty of provincial folks emigrated to the big metropolis of Rome, bringing their foreign ideas with them. Romans were always the sort to take up new fashions, so they had no problem incorporating some foreign beliefs into their daily rituals.
Egypt
The Egyptian goddess Isis was a particularly popular goddess in Rome. People identified her as Demeter, Selena (Roman goddess of the moon), and Hera. She fused with other mother goddesses and was worshiped as Diana on the Greek island of Crete. Romans equated Osiris with Dionysus and Horus with Apollo.
The King of the Egyptian gods was Amon, and the Greeks had long ago decided that Amon must be the same as Zeus, so they worshipped him, when they were in Egypt, as Zeus-Amon. After Alexander the Great conquered Egypt, the rulers of Egypt were all Greeks — a bunch of kings all named Ptolemy, and a long line of queens all named Cleopatra — and Egyptian religion became more Greeky (or Hellanized to use the technical term for "becoming more Greeky").
The last Cleopatra was the famous one, the one played by Elizabeth Taylor in the epic movie of the same name. After her, Egypt became a province of Rome, and Roman stuff got mixed in with the Greek stuff that was mixed in with the Egyptian stuff. (This happened whenever Romans moved into foreign countries – they encountered new deities and incorporated them into their mythologies.) So, where once there was a temple to plain-old Amon, by the first century AD there was one to Jupiter-Zeus-Amon!
When Christianity came to Rome and then to Egypt, things got even more complicated. Many folks who visit Egypt are surprised to see paintings that look like Mary and the Baby Jesus, but are actually Isis and Baby Horus. The two mother-and-divine-son stories were so similar that they influenced each other, and this shows up in art.
Persia
Before Rome became the huge empire that it was, a large portion of the world was under the control of the Persian Empire. This empire started in Iran (Persia), but spread through war and conquest until it stretched from India to the Aegean Sea from southern Russia to Ethiopia. Persians ruled Egypt for several hundred years, and also ruled many Greek cities in what is now Turkey. So, Persia was a big deal.
Alexander the Great, a Macedonian Greek, conquered the Persian Empire, but didn't put an end to Persian culture. He just replaced Persian bureaucrats with Macedonian bureaucrats, and things went on as they always had. Later, Romans replaced those Macedonian bureaucrats with Roman ones, but Persian culture still remained lively and was a rich part of much of the Roman Empire.
The Persians had two religions, which may or may not have overlapped:
- The Eastern Persians, the ones in Iran, were Zoroastrian. Their big god was Zarathustra.
- The big god of the Western Persians was Ahuramazda.
The Greeks and Romans came in contact with these religions, and thought they were interesting.
Plutarch — who was Greek by birth, but a Roman citizen — wrote a book about Isis and Osiris, two Egyptian gods, but included in it a bunch of stuff about Persian mythology. He talks about how Zoroastrianism was started by a priest named Zoroaster, 500 years before the Trojan War, who said that there were two gods fighting to control the world. One was good, either Zarathustra or Oromased, and the other was bad, Ahriman. The "umpire" in this context for control of the world was Mithra.
Mithra, even though he was Persian, was super popular with Romans, so much so that he became, essentially, a Roman god, under the name Mithras.
The most famous part of Persian religion today is the office of the Magi. Many people familiar with the Christian mythos know the Magi as the Three Wise Men from the East who followed a star to visit the Baby Jesus. Magi were Persian priests who specialized in interpreting dreams, reciting tales of the birth of the gods, and educating young rulers. These last two jobs of theirs would explain their trip to see little Jesus.