Although I'm not a historian, I do work as a miller on a historic windmill. What exactly would you like to know?
In short it seems the windmill has been invented independently in several places. Horizontal windmills have been built in Persia since about 600AD. These have then spread to China in the 13th century after Yelü Chucai observed them in Turkmenistan.
Europe however is a different matter. Water power has been used since antiquity to drive small mills with a horizontal water wheel.
In the first century BC the vertical watermill appeared. This type was described in detail by Roman engineer Vitruvius in 20 BC, who noted its angular transmission to drive the stones: revolutionary at the time and vital to the later windmill. Where the horizontal mill was limited to fast currents, this mill could work in slow rivers and had the potential to provide more power.
But after Vitruvius, despite its clear superiority to the horizontal mill, the vertical watermill seems to be forgotten. A reason for this could be the ample availability of slave labor for manual mills. The mill was again mentioned in the 5th century when slavery was abolished, after that there was a long silence in the dark ages. The watermills that were described during the dark ages were of such low yield they could only be of the horizontal type.
The vertical watermill reappeared in the 9th century in Flanders and from there spread rapidly over western Europe. Around the 11th century nearly all available power from streams and rivers was used for watermills in flat area's so other sources of power were sought.
This logically resulted in the first windmills, which are somewhat of a mystery. Depending on what historian you believe it is apparently invented somewhere in Flanders, Coastal France or England. The first hard textual evidence we have of windmills date from 1180 in Normandy, 1181 in England and 1191 in Flanders. All these texts mention already existing windmills.
What these windmills looked like is a matter of debate. The majority of historians agree they must be primitive versions of the post mill that still exists as a type today. The main problem with the wind in Western Europe is that it blows from different directions. The medieval solution to this was to place the mill on a central post on which it could be turned into the wind. There are few historians and millers who believe the early postmill was essentially a vertical watermill placed on a pole with the wheel swapped for sails. Although it would be a logical progression, there is no evidence for it save a 17th century sketch.
Originally these mills were quite small, and the post and crossplates were buried in the ground to prevent them from being blown over. The remains of buried crossplates have been found all over England and Flanders This subtype has the name 'sunk postmill'. Over the course of centuries this mill evolved -including a few critical changes improving its stability- into the full scale postmill in the mid-15th century that is completely above the ground. As a miller I'm still impressed with the amount of craftsmanship people had to build such a structure. Despite its rickety appearance, this final form of postmill is actually very stable. Many of them have survived for over 400 years.
In the 15th century the postmill was modified to be able to pump water out of area's that would otherwise be flooded, resulting in the 'Hollow Postmill'
Besides the wooden postmill, there is also the stone towermill. This particular type of mill is recorded not long after the first appearance of the postmill, in the 13th century. They originally were cylindrical buildings with very thick brickwork and a pointy tower cap. The towermill at Zeddam is a well preserved example. In medieval times they were nowhere nearly as popular as the postmill though for several reasons:
They were extremely expensive to make,
They could only drive one pair of stones where a postmill commonly had 2 or even 3
They required 2 millers to wind the cap where a postmill could be operated with 1.
As a result you see these medieval type towermills only in regions that were plagued with wars (a postmill would be too vulnerable) such as the nowadays border between Germany and The Netherlands.
Fast forward to the late-16th century with the invention of a less labor intensive method of winding, the invention of the 'main shaft'(allowing the mill to drive more than 1 pair of stones) and advancements in brickwork suddenly made the modernized towermill viable and they started to out compete postmills.
When stone was too heavy or expensive, a wooden octagonal structure was used, resulting in the smockmill
With the appearance of these modernized towermills and smockmills there suddenly was a lot of room for machinery and various transmissions which allowed people to use a mill for other functions than just grinding grain or pumping water. It was in the 17th century people started using mills for sawing wood, making paper, grinding spices, making paint and oil.
The first wind powered machine that is explicitly referenced was in a book by Heron of Alexandria in the first century AD. It was a sketch for a musical device powered by a wind turbine. However, it is unknown if this was actually produced, or if it was a theoretical design, or even if Heron created it. Link.
The next reference to a windmill comes from Islamic oral tradition, claiming that a Persian slave had the ability to produce a mill powered by wind sometime in the middle of the 7th century.
The first archaeological and written evidence for physical windmills is from 10th century Persia, where multiple writers talk about mills that use wind power to grind grain.
This is further corroborated by several detailed drawings from 1300 ad by separate sources of vertical windmills, Wulff, The Traditional Crafts of Persia, which functionally guarantees that windmills were definitely in existence by this point. However, they were relatively inefficient, did not make use of gearing, and were limited in scope. Yet, this remained the basic design of windmills up to the spread of horizontal windmills created by Europeans.
There is some disagreement on when windmills came to Europe, or if they were independently invented sometime in the 1100s, but there is archaeological evidence of horizontal windmills built around 1156. However, the European windmill was unique in that it used gearing and a horizontal axis for the mill, allowing significantly more efficient use of the wind and far greater sized mill stones due to the increased torque provided by gears. It is possible that the European design was a modified waterwheel, which would explain its divergence from the vertical mills present in Arabia.
From there, windmills were functionally unchanged for 800 years, aside from slight technological improvements such as bearings and better greasing mechanisms. They provided an efficient source of energy and one that was fairly reliable. They were used to grind grain, pump water, and drain lakes. As far as I can find, the only major addition to a horizontal design was to make it larger and add a base to take advantage of higher wind speeds at greater altitudes. This design rapidly spread and created the standard still present today.
TL;DR The Greeks thought it up, the Persians developed it first, and the Europeans may have adapted independently it from the waterwheel.
