Yes, the majority of New York City’s elementary schools are zoned, although there are also some non-zoned district schools, and city-wide schools. You can find your zoned school here.
You live in a school district that has dozens of “zoned schools.” One of those zoned schools is YOUR zoned school (i.e. where you have the highest priority). However, you are also given a lesser priority to schools within your district. YES, you can apply to schools in your district that are not your zoned school (you can also apply to some zoned schools outside of your district, but it is unlikely you will get a spot).
No, you can apply to any elementary school regardless of your address. You will be given a higher priority to your zoned school, and the majority of families do attend their zoned elementary school.
If you miss the application deadline, you should reach out to your family welcome center. You will be placed on waitlists at schools and be accepted based on availability.
There are no zoned middle schools in NYC. You can apply to any middle school in NYC, but have priority at middles within your district. You will be “matched” with one middle school based on your application choices, priority group for any given school, and randomized admissions (“lottery”) number.

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