China.
Chinese chillie sauces usually come as a thick paste, and are used either as a dipping sauce or in stir-frying.
Dou Ban Sauce (La Dou Ban Jiang 辣豆瓣醬, La Dou Jiang 辣豆醬, Dou ban jiang 豆瓣醬), ("la" is "spice", "dou" is "bean", "ban" is "piece", and "jiang" is "sauce") originates from Szechuan cuisine in which chillies are used liberally. It is made from broad or soy bean paste, and usually contain a fair amount of chillie. Often referred to in English as Chillie Bean Sauce.

Pao La Jiao, Yu La Jiao (泡辣椒, 鱼辣椒) Dipped Chillie or Fish Chillie. Made by pickling whole, fresh red chillies in a brine solution, this sauce is the key ingredient in the famous Sichuan dish YuXiang Rousi (鱼香肉丝),Julienned Pork in Fish Fragrance Sauce). The key to this pickle is to add a live crucian carp to the pickling pot along with the chillies, hence the name Fish Chillie. The carp is supposed to lend its fragrance and 'umami' to the pickle.

La Jiao You or Hong You (辣椒油, 红油) Chillie Oil or Red Oil, is another distinctive Sichuan flavoring found mainly in cold dishes as well as a few hot dishes. Chillie oil is made by pouring hot oil onto a bowl of dried chillies, to which some Sichuan pepper is usually added. After steeping in hot oil for at least a few hours, the oil takes on the taste and fragrance of chillie. The finer the chillie is ground, the stronger the flavor (regional preferences vary:ground chillie is usually used in Western China, while whole dried chillie is more common in Northern China.)

Guilin chillie sauce (Guìlín làjiāojiàng 桂林辣椒酱), made of fresh chillie, garlic and fermented soya beans. Also marketed as soy chillie sauce. (la jiao jiang and la dou ban jiang are not the same thing, though they look vaguely similar in the jar)

Duo Jiao sauce (Duo Jiao 剁椒) originates from Hunan cuisine, which is reputed to be even spicier than Sichuan cuisine. "Duo" means chopped, and "Jiao" means chillie. Duojiao is made of chopped red chillies pickled in a brine solution, and has a salty and sour pickled taste. Duo Jiao is the key flavoring in the signature Hunan dish Duo Jiao Yu Tou (剁椒鱼头), Fish Head steamed with Duo Jiao.


Korea
Gochujang

Vietnam:
Vietnamese hot sauce is made from sun-ripened chillie peppers, vinegar, garlic, sugar and salt. It is very popular in Vietnamese cuisine, often used in a wide variety of foods. It is commonly put on french fries in California[citation needed].

Thailand:
Thais put raw chillies on a very wide variety of food, in lieu of chillie sauces. They even put chillies on fresh cut fruit, such as watermelon. Some traditional foods have chillie sauces.
Thai sweet chillie sauce. Used as a dipping sauce. Mae Ploy is a leading manufacturer.
Nam prik, or Thai chillie sauce. Fish sauce (Nam Plaa) with raw chillies floating in it.
Sriracha sauce is a Thai style of chillie sauce.

Japan:
Rayu or La Yu Chillie Oil (辣油, Chinese 辣椒油), same as La Jiao You. Often used for dishes such as gyoza.
Shichimi Togarashi (七味唐辛子)or Ichimi Togarashi (一味唐辛子)are seven or one ingredient spicy seasoning mixes, with chillie. Used for many soupy foods, such as udon.
Okinawa - Kōrēgūsu (コーレーグース, 高麗胡椒), made of chillies infused in awamori rice spirit, is a popular condiment to Okinawan dishes such as Okinawa soba. It refers to Goguryeo.

Malaysia, Indonesia, Brunei, Singapore:
sos cili, similar to Thai sweet sauce.