
The history of the art of horse riding dates back almost 2,500 years to the time of the Greeks. The history of the Spanish Riding School in Vienna to the 16th century.
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Greeks and Spaniards (?)

(View down the cloister running along outside the stables of the Spanish Riding School)
Xenophon — a student of Socrates — wrote the first (and, for a long time, only) books on the subject of horse riding: On Horsemanship and The Cavalry General.
Horsemanship continued to flourish under the Romans, but the so-called dark ages and the rather simple demands placed on medieval warhorses — presumably “run as fast as you can at the enemy” — did little for the development of classical horse-riding skills.

(Part of an equestrian portrait of Emperor Karl VI, who reigned when the new Spanish Riding School formally opened. Image courtesy of the Rijksmuseum)
Fortunately, equestrianism experienced a resurgence in the mid-1500s, led in part by the need for more intelligent use of the horse in battle given that the riders now wielded firearms.
That resurgence eventually reached Vienna, where documents dating back to 1572 already refer to a wooden Spanish riding hall.
By the late 17th century, the court decided it needed a more permanent and sturdy riding environment than the existing wooden hall could provide.
As usual in days past, war interfered with imperial plans. So it was only in 1729 that work on today’s Spanish Riding School (German: Spanische Hofreitschule) began within the buildings that form the Hofburg Palace complex.
The enclosed arena is largely unchanged since its official opening in 1735 under the aegis of Emperor Karl VI (whose portrait hangs at one end of the hall).

(View of the new 1735 Imperial riding school; drawn by Salomon Kleiner, engraved by Johann Bernhard Hattinger, and published by Johann Andreas d. Ä. Pfeffel in 1737; Wien Museum Inv.-Nr. 105765/108; excerpt reproduced with permission under the terms of the CC0 licence)
The architect behind the new design was one Johann Bernhard Fischer von Erlach, whose pen and ruler found further use in two other notable Baroque buildings from Karl VI’s reign; the Karlskirche church and the Prunksaal of the national library still belong to the very best of Vienna’s architectural jewels.
White stone balustrades, galleries, and columns dominate the architecture of the riding arena, with soft lighting from mammoth chandeliers.
The hall provides a majestic environment for the world’s oldest working riding school still practicing the classical art of renaissance dressage.

(The imperial carousel in the winter riding school on November 23rd, 1814; published by Artaria & Co. Verlag; Wien Museum Inv.-Nr. 23711/1, excerpt reproduced with permission under the terms of the CC0 licence)
Although synonymous with the Spanish Riding School, the arena building is actually just the winter school, where the performances and public trainings take place.
There are stables over the road, for example, which are not normally accessible to the public outside of their guided tours. Plus a stud farm in the province of Styria and a training centre in the country at Heldenberg in Lower Austria.
P.S. Why Spanish?
It seems Spanish horse breeds were best suited to the equestrian arts, and the Habsburg court used them exclusively for that purpose. The tradition continues today at the modern riding school.
Not just any Spanish horse finds its way into the Stallburg stables in Vienna, mind you…only the Lipizzaner, a breed which dates back to 1580 and Karl II Franz’s stud farm (see the article on the Lipizzaner stallions).