Africa

It is impossible to leave Africa without mentioning Egyptian horses and the high-class desert Arabians, bred for many years in the good climatic conditions of the El Zahraa and Ein Shams studs.
For lack of a country to put the Zebra under, I am providing the link here for now.

The Empire of Equus lists the following breeds under Africa; I have left the chart as is for now until I can sort it out better under modern countries.

Abyssinia: Dongala horse, Galla horse (Arab/Barb cross)
Algeria: Arabian, Barb, Dongala horse
Cameroons: Fullah horse, Lakka pony
Central Africa: Togo pony
Congo (Zaire): (no domestic horses, because of the tsetse fly)
Madagascar: Madagascar horse; horses from Abyssinia, Algeria, Arabia, and Tarbes (Spain)
Morocco: Arabian, Barb, Dongala horse, Libyan Leopard horse, and imported Breton, Percheron, and Boulonnaise draft horses
Nile Region (upper): Dar-Fur horse, Dongala horse, Kordofan horse
Senegal and Gambia: Bagazan or Kinabuta horse, Dongala horse, Arab-Barb crossbreeds
Somaliland: Galla horse, Somali horse
South Africa: African horse, Cape horse, Basuto pony
Southwest Africa: Native S.W. African horse; imported East Prussian, Gidran, English Thoroughbred, and other British breeds
Sudan: Sudanese horse
Togoland (West Africa): Descendants of Barb horses; Togo pony
Tripolitania (Libya): Arabian, Barb, and Arab/Barb crosses
Tunisia: Arabian, Barb, Beradin horse, "Bou Chareb," English Thoroughbred, mountain pony

Morocco

Asia

Horses are valuable in the Lebanon and Beirut, and many are part or true Thoroughbreds, or Anglo-Arab types, but there are "asil" horses too, as there are in Iraq.  Sometimes a horse shows all the Arabian characteristics, and in fact traces back many years to an Iraqi stallion, Ahsuri--a Thoroughbred that was disqualified in Beirut as a non-Arabian, before being taken to Iraq and crossed with pure Arab mares.
The strong, often dun-colored ponies of Tibet are descendants of domesticated Mongolian and Chinese ponies.  Wealthy Tibetans, including the Dalai Lhama, used to own small studs--which are almost certainly dispersed by now.
In addition to much that I have already placed under the various countries, The Empire of Equus lists the following:  Afghanistan as the home of the Afghan horse, Kabul horse, and Kandahar horse; Indochina as the home of Cambodian, Burmese, South-Annam, Tonking and other native horses, as well as various imported breeds; Iraq as the home of the Karabac horse (related to the Karabakh or different?); the Philippines as the home of the Sulu horse (from Sumatra) and the Philippine pony (various types); and Syria as the home of the Anaze horse.

Azerbaijan

China

India

Indonesia

Iran

Japan

Jordan

Kazakhstan

Korea

Latvia

Lithuania

Mongolia

New Zealand

Persia

Russia

Saudi Arabia

Tajikistan

Turkey

Turkmenistan

Uzbekistan

Horses were unknown on this continent until introduced by the first explorers from Spain or Holland in the 16th century.  With the arrival of the British convict settlements and the opening up of the land to cattle and sheep, horses became a necessity.
Today the units of mounted police, still found in most states, have shifted from their omnifarious, indispensable work in the outback, to traffic and crowd control in the cities.  Instead of the heavy, weight-carrying animals of the trooper police, they ride quality horses, well-suited to public exhibitions of horsemanship and their many ceremonial duties.
Australians enthusiastically support all forms of racing, from the "picnic races" of the outback to the celebrated Melbourne Cup, taking place each autumn.  There are a growing number of first-class Thoroughbred studs, many blood animals being used not only for racing but for mustering on sheep or cattle stations, and even as pony club mounts.  Numerous Arabian horses are bred, as well as European breeds.
Quarter horses were first imported in 1954.  In 1967, Mr. Lougher set out from America with a shipment of thirty, to found another Australian stud.  Because of hold-ups due to swamp fever regulations, his voyage eventually took 13 months and hit the world headlines.  These horses are increasingly popular as stockhorses.
The Empire of Equus lists the following breeds under Australia:  Cross-bred Arabian, Indian, and Persian horses; English Thoroughbred; Timor pony; and "Waler" (an Anglo-Indian horse bred in New South Wales and exported to India for cavalry service).

I don't really know where the Gypsy Vanner originated, but I think it was somewhere in Europe so I'm putting the link here for now.  Also, until I get a page for Estonia, I will just mention here that it is or was apparently the home to such breeds as the Unveredeltes farm horse, the Torgelscher strain, the Smudish horse, and the Klepper, as well as crosses between farm horses and the Ardennais draft.  I must also mention that Danish and Friesian drafts and East Prussian (Trakehner) horses can be found in Latvia; the Zemaitukas pony and the Pange horse are bred in Lithuania; and Rumania (Romania) is the home of the Bulgarian or South-Bessarabian horse, the German-Bessarabian horse, the Huzulen horse, the Rumanian farm horse (4 strains), the Dobrutscha horse, a mountain horse, the Moldavian horse, the Transylvanian horse ("Seven cities" and Jalomitz strains), and numerous imported breeds.
The Zemaitukas is an ancient breed of large ponies or small horses of Lithuania, typically dun in color with light mane and tail and a dark mid-dorsal stripe.  Mouse-grey and bay colors also occur.  Physically the Zemaitukas is a well-documented breed.  The average height is 56.6 inches for stallions and 55 inches for mares; the average weight, 893 pounds for stallions and 872 pounds for mares.  Average birth weights are 84 pounds for male foals and 82 pounds for females.

Austria

Belgium

Bulgaria

Czechoslovakia

Denmark

England

Finland

France

Germany

Great Britain

Greece

Holland

Hungary

Iceland

Ireland

Italy

Netherlands

Norway

Poland

Portugal

Russia

Scotland

Spain

Sweden

Switzerland

United Kingdom

Wales

Yugoslavia

Horses like the Mustang, the Appaloosa, and the Paint are often put in the catch-all category of "Indian ponies."  Their story is a fascinating chapter in the history of the horse in America.  It begins with the arrival in the New World of the Spanish explorers.  The horses they brought with them were perhaps the finest of their time and the first to walk the continent since Equus, the ancestor of all horses, disappeared from it.
Over the centuries breeders on the Iberian peninsula had produced excellent results from available strains.  As early as the fifth century B.C., an important breed called the Vilana had been developed by crossing native stock with the cold-blooded Northern horses of Celt invaders.  In the third century BC, the Vilanas were bred to Eastern stallions brought by the army of Carthage on its roundabout way to Rome.  A horse of extraordinary speed and substance resulted.  Many centuries later, when the Moors conquered Spain, they brought their culture and their splendid desert horses with them.  With this infusion of Eastern blood, the Spanish horses moved another step closer to perfection.  The mounts of the Spanish conquistadors and explorers were mostly Andalusian-bred and strong in the blood of the Barbary (Moroccan) horse.  Thus the type of horse that debarked on the shores of the New World was in general very fine, the result of centuries of intelligent breeding.  These were the horses--whether strayed, stolen, or bartered--that came into the possession of the American Indians.  The "Indian pony," product of indiscriminate mating and the harsh plains environment, is their direct descendant.
The Indians took little time to get over their initial awe of the horse.  During the 17th century many of the Southwestern tribes built up sizable herds and their braves learned to ride with fantastic skill.  The horse provided greater range for tribal bands; it allowed hunters to kill single buffalo instead of stampeding herds into pit traps; and it made the Indian brave armed with bow and arrow fearsome in war.
The Cayuse tribe also bred good horses.  Environment was a factor, since both the Cayuse and the Nez Perce tribes lived in a mountainous region where the animals had good forage.  On the Plains, the Indian ponies tended to be, as the artist George Catlin noted of the Comanche herd, "generally small, of the wild breed, tough and serviceable."  Unlike the Cayuse and Nez Perce tribes, which selected both mares and stallions and gelded unsuitable males, the Plains tribes usually allowed their horses to mate freely.
Horsemen of today are trying to classify and improve the distinct types of Western horses by crossing with breeds of highly uniform and desirable characteristics, particularly the Thoroughbred and Arabian.  For the present it is the coloration of the Appaloosa, the Pinto, and the Palomino that differentiates them.  Conformation within each breed is still highly variable, but in time standards will be established.

Canada

Mexico

Puerto Rico

United States

The conquistadors and other pioneers brought horses and a common quality of fine horsemanship which was developed even further by the cattle riders of early Spanish colonial days.  Because these first Latin American cattle riders were immeasurably tough and superlative horsemen who could ride anything, they were exceptionally fitted for conquering new territory, establishing cattle ranches and providing ready-made, first-class irregular cavalry.
South America is horse country, although breeding is difficult in a few regions because of disease.  Peru is also ideal country for raising fine types, many of which spread into the other republics.  Basically, the majority of South American horses are variations of the Criollo.  Terrain and climate produce their own changes.
In South America, Belgian and Flemish horses, as well as Percherons, have been imported from time to time, although heavy horses do not stand up well to constant heat and flies, and are peculiarly susceptible to ticks.
The Venezuelan Llaneros ride a lighter, taller type of Criollo with finer limbs, which is named after them--the Llanero, or Prairie Horse.  The Chilean type, working in the mountains and usually under great hardship, was noted more for strength than beauty, but has been much improved by careful breeding in the last twenty years or so.  It is hoped they retain their sensible ability to grow profuse manes and tails as protection against the flies.
The Brazilian Crioulo is little different from its cousin in Peru, the Criollo or Costeno and the Marochuco, of more angular conformation, found in the mountain regions.  Both these working horses are descended from those brought to Brazil by Pizarro in 1532.
South America is indeed a land of horses.  In Brazil they say, "God first made man.  He thought better of it and made woman.  When He got time He made the horse, which has the courage and spirit of man and the grace and beauty of woman."
The Empire of Equus lists the following breeds found in South America:  Criollo, Mestizo (half-bred), Anglo-Argentine cross-breds, Chilian pony, Peruvian Paso, Morochuco (curly-coated pony), Bagual (feral) horse, and numerous imported breeds including the Arab, Anglo-Normand, Polo pony, Thoroughbred, Percheron, Clydesdale, Shire, Suffolk, Boulonnais, Hunter, Hackney, Standardbred, Belgian, Norman, Norfolk, Shetland pony, Yorkshire, Ardennais, Orloff trotter, Welsh pony, Breton, Oldenburg, Holstein, and others.
The Peruvian Paso is a riding horse bred in Peru since the early 1500s, but not imported in numbers into the United States until 1960.