You can buy deodorant in Japan (see the link Garrick posted in the comments). It's called デオドラント (deodoranto). In terms of personal care items though, it's a second-tier product, by which I

mean it's not ubiquitous like soap or something, so not all stores may stock it, or they may only stock a couple of brands, depending on where you are. If you're looking for specific American brands you may have to go to specific stores, or buy them by mailorder. Amazon and Rakuten both stock deodorants in abundance. See デオドラント・制汗剤 - パーソナルケア: ヘルス&ビューティー for the Amazon search page result for デオドラント. Here's the search result page for "American deodorant" on Rakuten: 【楽天市場】アメリカ製 デオドラント の検索結果 - (標準順 ウィンドウショッピング):通販・インターネットショッピング

Deodorants as a product category have never really caught on in a huge way in Japan, probably for a couple of reasons. First, there's a perception that it's meant to cover up the odor rather than eliminate it, which goes rather against the idea that odor and dirt should be washed off frequently. On Makiko Itoh's answer to Japanese Culture: What are some of the Japanese hygiene habits which the rest of the world should know?, I describe a soap that is supposed to deal with 加齢臭 (kareishuu) or the smell of getting older. Or on the Amazon search page I linked to above, the 3rd product listed is a special anti-odor foot soap. That is the way body odor is usually tackled by many Japanese people.

Another reason is, as far as anti-perspirants go, when faced with the typical sweltering summer that is experienced in most of Japan, chemical ones are pretty useless. (There are also concerns about how healthy they are since they clog up the pores.) So people carry around handkerchiefs all the time with which to wipe their sweat off. Some women use anti-perspirant pads, like this one.

There are anti-perspirant pads for the nape of the neck and so on for kids, especially ones with sensitive skin. Babies are very prone to getting prickly heat, although everyone gets it sooner or later. (Anecdotally, I only get prickly heat when I'm in Japan in the summer. It's that almost saturation-point humidity.)

There are products like this too - anti-odor undershirts - although most men just change their undershirts frequently in the summer. (And undershirts are a must, otherwise your sweat just soaks through to your outer shirt.)

One more reason: traditionally, Japanese people have thought that they don't have that much body odor because they don't eat a lot of meat or spicy food. Conversely, there's a prejudiced view of westerners as being stinky because they eat a lot of meat and dairy products. An old fashioned way to say someone was "too foreign" was to say they 'smelled like butter'. (See: Stinks like butter! by Makiko Itoh on Japan - it's so weird and wacky! ) You can find serious arguments online opining that Japanese people are getting stinkier because their diet is more westernized. I have no idea if there's any scientific correlation between meat in the diet and body odor.

The body-cooling products are pretty new. It's to be seen if they really catch on. The idea sounds nice though.

Also, as I was researching this answer I came upon this: 日本にはデオドラント用品がなぜこんなに多いですか? - Y!知恵袋 on the Japanese equivalent of Yahoo! Answers, which asks, "why are there so many deodorant products in Japan compared to other Asian countries?" So I guess it's a matter of perspective.

More about Japanese hygiene habits: Makiko Itoh's answer to What are some of the Japanese hygiene habits which the rest of the world should know?

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