I often joke about my characters having superpowers. None of them are superheroes - what I mean by this is that they have special skills, or special powers of intuition, that allow them to get through the story. It makes sense to do this - if your protagonist is going to beat the bad guy, likely enough he or she will need to be a good sword fighter, or an expert thief, or able to put two and two together at critical points in the story.
First off, there's the danger of employing Superman in your story. You don't want your protagonist to be so powerful that nothing can hurt him. Vulnerabilities and self-doubt can go a long way toward making a protagonist relatable. It may be a product of our modern age that people are less interested in seeing an uncomplicated hero - on the other hand, look at Hamlet. He had issues, too (not that they were called "issues"!)
The other danger is that you can't give people sudden unexplained skills - because then they will be superpowers. In The Lion, the Witch, and the Wardrobe, Peter gets handed a sword and suddenly knows how to use it - something I always ascribed to magic in the sword, but I've had friends tell me that it always bothered them.
This is something where backstory becomes critical. Give your characters a chance to learn the things they need in order to be who they are. What that background needs to be will depend on the character and the power or skill. Sometimes it will be a process that must be shown in the story so that people can see the characters developing their abilities. Sometimes it can be alluded to in backstory that this character went to a special school, or had a martial arts mentor, etc. etc. To that you can then add whatever natural intuitions etc. the character has.
Let's look at some examples.
Ender from Orson Scott Card's Ender's Game is a genius kid. I'm sure you've seen more than a few genius kids in your reading history. Ender has the advantage that the author has specified that he's part of a deliberate breeding program that is trying to create genius kids. The idea that a genius kid would arise spontaneously (with two more genius kids as his brother and sister) would stretch credulity. That there could be a deliberate program trying to achieve this, and that they could then achieve this in different ways in the same family, is much less improbable. Much easier to accept as a science-fictional conceit.
Another character who is gifted with both natural characteristics and training is Phèdre from Kushiel's Dart by Jacqueline Carey. Her supernatural ability is her tolerance/enjoyment of pain, and her training comes from her long years of specialized education in Anafiel Delaunay's household.
My own character, Xinta, from "The Eminence's Match," is a bodyguard and a political assistant as well as a personal valet. When I first wrote him (many, many years ago!) I noticed he was coming across as graceful, polite, incredibly dexterous, awesomely powerful in martial arts, etc., and all of it was completely unjustifiable. He was a superhero. Since I didn't want superheroes in my story, I turned to education. In the complete model of my world all manservants are intensively educated from age 6 at a special academy where individuals who don't possess the mental or physical qualifications are slowly weeded out and sent into other lines of work. The ones who succeed are trained in various academic subjects, manual dexterity as well as bodyguarding and (for those serving men) how to read people's faces. Their manners come from their upbringing within the culture of the servant caste. My character Aloran from For Love, For Power has been trained to work for ladies, and that includes special training in medical care. To that I can add an edge of intuition that serves him well in the story, but isn't so unusual that it could be considered supernatural.
When I'm putting together a story, I often find it helpful to think in terms of superpowers. It's my playful way of talking about the special skills and qualities each major character brings to the story. It also helps me keep these qualities balanced between people, so that everyone can contribute.
It's something to think about.