I'm learning how to write hiragana, katakana and kanji.
What is the name of paper that has little squares to write the Japanese characters?
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This kind of paper is called 原稿用紙, used in composition class or by professional writers, but for practical writing rather than training (generally too small for training).
The format below is dedicated to kanji training (but also good for kana). While it doesn't have fixed name, notebooks that bind these sheets are sold in the name of 漢字練習帳.