There are public schools for gifted children in Russia, and I’d guess in many places. The usual setup is not to filter directly by IQ, as Viliam_Bur describes (which does seem potentially socially inappropriate) but to admit students based on hard competitive tests in math/physics, and have as teachers either college professors or former math/physics olympians.
Filtering by mathematics correlates positively with filtering by IQ, but it is not the same thing. Most gifted people are not great in maths.
One of advantages of teaching in a school of children filtered specifically by IQ was that it disproved my many prejudices about high-IQ people. Those children were different from the average, but it was not easy to say how exactly; the simple explanations all failed. For example some of them were great at maths, certainly much higher ratio than in the average school, but most of them were not. Similarly with other traits—many interesting traits were more frequent than in the normal population, and yet, even within the gifted population they remained a minority; just a greater minority than usual. If you try to filter by these traits, you will find many gifted people, but even more gifted people you will filter out.
As a student I was in a math-oriented high school. I would say most people there were highly intelligent too. Yet, it was different. The interesting aspect of the IQ-filtering school for gifted children is that you get many talents of a different kind together. Often you have one child with multiple talents—in a math-oriented school they would probably develop some of them and supress the rest. In IQ-oriented school, they can develop their math talent during the math lessons, and e.g. artistic talent during their art lessons. Without an IQ-oriented school they would have to choose, either math or art; or at least only one of them at school, and other only in afternoon activities.
As an example: on the school for gifted children we started organizing a high-school competition in making computer games. That is not exceptional per se; there exist many other programming competitions. The difference is that in a typical programming competition, you have an exact problem, and you get points for algorithm that solves the problem. In this game-making competition you get points for multiple aspects of the game: graphics, music, storyline, presentation, and the technical flawlessness. So the programming is important, but the algorithm is not everything, other aspects matter too. In my opinion, this difference is a nice metaphor for the difference between the schools.
Filtering by mathematics correlates positively with filtering by IQ, but it is not the same thing. Most gifted people are not great in maths.
One of advantages of teaching in a school of children filtered specifically by IQ was that it disproved my many prejudices about high-IQ people. Those children were different from the average, but it was not easy to say how exactly; the simple explanations all failed. For example some of them were great at maths, certainly much higher ratio than in the average school, but most of them were not. Similarly with other traits—many interesting traits were more frequent than in the normal population, and yet, even within the gifted population they remained a minority; just a greater minority than usual. If you try to filter by these traits, you will find many gifted people, but even more gifted people you will filter out.
As a student I was in a math-oriented high school. I would say most people there were highly intelligent too. Yet, it was different. The interesting aspect of the IQ-filtering school for gifted children is that you get many talents of a different kind together. Often you have one child with multiple talents—in a math-oriented school they would probably develop some of them and supress the rest. In IQ-oriented school, they can develop their math talent during the math lessons, and e.g. artistic talent during their art lessons. Without an IQ-oriented school they would have to choose, either math or art; or at least only one of them at school, and other only in afternoon activities.
As an example: on the school for gifted children we started organizing a high-school competition in making computer games. That is not exceptional per se; there exist many other programming competitions. The difference is that in a typical programming competition, you have an exact problem, and you get points for algorithm that solves the problem. In this game-making competition you get points for multiple aspects of the game: graphics, music, storyline, presentation, and the technical flawlessness. So the programming is important, but the algorithm is not everything, other aspects matter too. In my opinion, this difference is a nice metaphor for the difference between the schools.