Demon Slayer's breath styles (呼吸, kokyū) are a fictional martial system in which the swordsman uses a specific breathing pattern to oxygenate the blood beyond normal human capacity. The increased oxygen unlocks superhuman speed, strength, and pain tolerance, with the breathing pattern shaping the sword forms that the user can perform. Flame Breathing (炎の呼吸) is one of the five oldest branches, descended directly from Sun Breathing (日の呼吸) — the original style of the first demon slayer Yoriichi Tsugikuni.

Flame Breathing has nine forms (型). They are titled in the classical Japanese pattern that names martial forms — number + name, often using older or literary kanji — for example 壱ノ型 「不知火」 (Form 1: Unknown Fire), 弐ノ型 「昇り炎天」 (Form 2: Rising Scorching Sun), 玖ノ型 「煉獄」 (Form 9: Rengoku, 'Purgatorial Flame'). The ninth form shares its name with Kyojuro's family name — that pun is not an accident, it tracks back to a Rengoku family ancestor whose name became the technique's name. The forms emphasise broad arcing strikes and high-temperature nichirin blade coloration; in the anime adaptation the visual signature is the flaming dragon that wraps around the swordsman during a full extension.

Flame Breathing is hereditary in the Rengoku family — Kyojuro's father Shinjuro is the eighth-generation user, Kyojuro is the ninth — and the Flame Hashira seat has historically been a Rengoku seat. The lineage is the same engine that drives the family's pride and its grief.

The form-name 玖ノ型 (Form 9) uses 玖 instead of 九 for 'nine' — an old, ceremonial variant used on documents and in tea-ceremony numerals. The author uses this older numeral specifically to signal 'classical / hereditary / not casual modernity,' the same flavour that animates Rengoku's speech register.

The forms, form by form

• First Form — Shiranui (Un-Knowing Fire) — When a technique name is also a ghost story.
• Second Form — Rising Scorching Sun — How English caught the sky.
• Fourth Form — Blooming Flame Undulation — Lyric in kata.