Some of these whiz kids shine in the classroom, some in the music room, some on the sports field. A few have overcome daunting handicaps that make their accomplishments all the more surprising, and inspiring.
But first, a look at two young prodigies in the sports world.
In 1997, 21-year-old Tiger Woods became the youngest golfer ever to win the Masters Tournament.
Today, there's an up-and-coming teenager, Michelle Wie, who's set her sights on the Masters.
But for Wie, that's not enough. She also wants to compete with men, and beat them.
When Correspondent Steve Kroft met Wie last spring, she was on a roll, and she was barely 14 years old.
"I think my ultimate goal is to play in the Masters," she says. "I think it would be pretty neat, walking down the Masters' Fairway."
Does she think the world is ready for it? "Yes, I think they are. I think they are getting ready for it."
Lots of 14-year-olds dream about winning the Masters, but this is not a fantasy for Wie. A year ago, she became the youngest person to ever take on the men in a PGA tour event. And in front of a hometown crowd in Honolulu at the Sony Open, she performed like a debutante at her coming-out party.
She shot 72-68, even par, missing the 36-hole cut by a single stroke. But it was still good enough to tie or beat 64 of the best men golfers in the world, making a believer out of skeptical sports writers like John Hawkins of Golf World.
Where is she in her career now, compared to where Woods was at that age?
"Taking gender out of the equation? She's ahead of Tiger," says Hawkins. "She could do more for golf than Tiger Woods."
At 10, Wie had already shot a 64, and challenged some of the top women amateurs at the U.S. Public Links Championship. Last summer, she won the tournament at 13, her first national title, and qualified for five out of the six events she played in on the ladies professional golf tour.
"I like challenges," says Wie. "And I just think that the faster you get there, then it seems like you always have, I always have to be the first to do everything, and I just want to be the best."
Wie is just over 6 feet tall, has a perfect swing, and already drives the ball about as far as the average golfer on the men's tour. She is so accomplished and so polished that it's easy to forget that she is just a ninth grader at Punahou High.
She watches "American Idol," listens to Coldplay and Good Charlotte, loves Jim Carrey movies, and goes to the mall with her friends. She also works out every day, practices for three hours after school, and eight hours on the weekend. Other than that, she seems very much the normal teenager with all the normal anxieties.
Does she feel different because of all this? "I guess I'm different because I'm, well, not just because of golf. I guess I'm different because I'm just taller than everyone else. I'm just freakishly tall," says Wie. "I really don't want to grow any more."
She began her golf career at 4, and was introduced to the game by her parents, who both immigrated to the United States from Korea. Her mother, Bo, now a Honolulu realtor, had been an amateur champion there. Her father, BJ, a professor at the University of Hawaii, has been a 2 handicapper, which means he is very good.
Wie says she start beating her parents at golf at the age of 7 or 8. "Well, they say I started beating them when I was 9, but I refused to believe that," recalls Wie, who says her mother pushed her more in the game. "I wouldn't really call it pushing, because they didn't really push me that much. It was more self-motivated."
Her game is now in the hands of one of the best instructors in the business, David Leadbetter, who has coached the likes of Ernie Els and Nick Price.
Could Wie make a living on the men's tour?
"If she continues at the present rate, I'd say yes," says Leadbetter. "You'd have to say that potentially she could do it, and I wouldn't put it past her. I mean, she's a very competitive girl."
Whether performing for 60 Minutes cameras, or surrounded by huge galleries in front of a national television audience, Wie has already demonstrated the one quality necessary to be a champion, and that's the ability to elevate her game to another level when she has to.
"She's a gamer; she's a performer. Michelle loves the stage. She's got a lot of personality. She's got nothing to worry about, except when the train gets moving too fast," says Hawkins.
"Michelle's a great player. But she's 14 and she's female. And she would not be able to compete on anything close to a consistent basis with the best men's players in the world -- best male players in the world."
He says he also doesn't approve of the fact that she's playing in PGA tour events: "I blame that as much on the PGA Tour as I do on BJ and Michelle Wie."
"I get really bored easily, so if I just play in the women's tournaments and if I just play them over and over again, I think I'll get bored of golf," says Wie.
What does she say to people who think there's no way a woman will be able to compete with men in professional golf? "Well, I think the reason they're saying that is because they're truly afraid," says Wie. "I mean, men's egos can be easily brought down, and I don't think they want that to happen."
She knows this at 14? "I have some experience with my dad, and I think that it's just the way guys are," she says, with the confidence of someone who has been beating boys and full-grown men all of her life.
"You could turn pro tomorrow and be rich and famous," says Kroft.
"Yeah, but I think I'd like to go through the basic steps of life, like go to high school, and then go to college and be in a dorm and stuff like that," says Wie. "I think I just want to go through the basic steps of life, and then I'll be fine from then."
Wie, now 15 and in 10th grade, didn't qualify for this year's Sony Open, but she has a busy schedule on the ladies' golf tour.
Ten months ago, sports prodigy Freddy Adu made his debut as a professional soccer player, pulling down a record salary of $500,000 a year.
That's not bad for a 14-year-old. But as Correspondent Lesley Stahl reports, Adu has been offered even bigger money.
Adu can do just about anything he wants with a soccer ball, including fake much older players right out of their cleats. People who know say that he could develop into the best player in America -- and in the entire soccer-mad world.
"I just love it so much, you know? When I'm out there on the field, I'm in a whole different world," says Adu. "It's like, I'm just having so much fun."
And he gets paid a lot, too, to just have fun. "I couldn't ask for a better life," he says.
Adu does seem to have a charmed life. He get half a million dollars a year to play for DC United, the team he grew up watching. At an age when most kids are begging their parents for a higher allowance, he's got a million-dollar contract with Nike, and another endorsement deal with Pepsi.
"We think he's going to be a great one," says Bruce Arena, the most influential figure in American soccer. He's also coach of the men's national team that will compete for the World Cup in 2006.
It's a team that Adu hopes to make.
"He's strong, he's quick, he's agile," says Arena. "He's got good balance and he's got great vision. Very special."
Adu learned the game as soon as he learned to walk in his native Ghana, in West Africa. "There were no coaches, no one to tell you what to do. It was just, you play and learn stuff on your own," he says.
His mother, Emelia, says Adu was always holding a soccer ball. And Adu says it was during those early days in Ghana when his mother first encouraged her soccer prodigy: "My mother was always the supplier of soccer balls. And so people were always knocking on my door trying to get me out so we could play."
Adu's street soccer days ended in 1998, after his parents entered a visa lottery at the U.S. embassy in Ghana. He was 8 years old. "I didn't even have a ball when I first came," recalls Adu. "I didn't even have a soccer ball, so it was hard."
But life got even harder. After they settled near Washington, D.C., Adu says his father just abandoned the family one day. "I don't see him," he says. "He's completely out of the picture now."
Emelia suddenly became the sole breadwinner for Adu and his younger brother, and worked two jobs to make ends meet.
"She would wake up at 5 in the morning, leave at 6, goes to work. She's done, she gets off work at 6, you know, in the evening to go to work overnight job," recalls Adu. "She worked, I mean, unbelievable amount of hours."
Meanwhile, Adu's soccer skills were noticed on the playground, and he was recruited for a local league team. In less than a year, he was the best player on an all-star team, playing with kids four and five years older than he was.
"One of the big soccer teams in Europe, actually an Italian team, didn't they offer you a big, huge, amount of money to come and play for them," asks Stahl.
"Yeah," says Adu. "It was $750,000 to play for Inter Milan."
At the time, Adu was just 10, and his mother said no. Was he upset? "No," he says. "But it was just so much money. Why not just take it? But you know what? She was looking out for our well-being."
Eventually, his mother did agree to let Adu attend a kind of soccer boarding school in Florida, run by the U.S. Soccer Federation. He became a citizen, and officially joined the United States under-17 team. But before his professional debut, Adu's mom insisted on one more thing -- that he complete an accelerated academic program for athletes in Florida that allowed him to finish high school at 14.
Are there any ways in which this kid is still able to be a kid?
"I play a lot of Playstation, and always trying to look pretty for the girls," says Adu.
But there are expectations being heaped on him. You hear that he's the next Pele, that he'll be the youngest player ever in a World Cup, and that he'll finally make America as mad about soccer as the rest of the world is right now.
On draft day, Adu, the No. 1 pick, may not have measured up to major league soccer's other top rookies. But his $500,000 salary dwarfs even the veterans.
It's become a cliché that every rookie with a big contract buys his mother a house. But Adu may be the first one ever to do it, and then go home and live with her.
"I wanted to get her a place, a home with a little bit more space," says Adu. "The kitchen is just huge, because my mom is, she lives there, man. She loves being in the kitchen."
Is she a good cook? "Oh my goodness. She could be the best cook ever, man," says Adu.
He adds that his mother doesn't work any more: "She's done. She's had enough. You know, she's worked so hard."
"What I know of Mrs. Adu, she's pretty sharp," says Arena. "And she's done a very good job in corralling Freddy a bit, and not letting his head get too big."
So how's Adu going to deal with the spotlight, and the pressure that will come on the soccer field? "They say that there's a bulls-eye on your back sometimes when you play," says Arena. "He's going to have one on his back, and on his chest, on his forehead, and on his legs."
"I don't have to prove myself to any of those guys that, that I'm better than them," says Adu. "I'm just going to go out there and play, man."
How's he going to get to practice? He's 14, and he doesn't drive. "Actually, I don't know," says Adu. "My mom is probably going to have to take me back and forth."
And that's exactly what happened. Adu's mother drives him to practice, and she says it's the only quiet time they have together. Now 15, Adu had a good first year with DC United. He played in every match during the regular season, and the team won the Major League Soccer championship.