There seems to be some dispute about which was the very first instance of a wristwatch, but the first writstwatch appears to have been made by Patek Philippe in 1968 for Countess Koscowicz of Hungary. Wrist watches remained somewhat in obscurity, but two key events are often mentioned as key in their history. In 1904 Louis Cartier created the first wristwatch for aviator Santos Dumont who wanted to keep his hands free while flying. It was certainly a notable moment in wristwatch history, but not the first. The other key event was World War I. It was during the First World War when their inherent functional advantages over the pocket watch seemed to increase their popularity.

While it's not what we may see as a "wristwatch" it had the exact same purpose and functionality. The sundial ring was used as early as the 17th century. It was worn on the wrist or carried on the person and removed to tell the time. When held correctly, the small hole on the top would shine through into the inner circle of the ring and shine upon the time.

Patek Phillipe invented the wrist watch in the late 1800's. I never heard of an argument about it.

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The invention of the watch began in Europe in the 1500's.
In 1675, England paved the way for the invention of the watch by creating the hairspring..England led the watchmaking revolution, there were important contributions in the development of mechanics and the elegance were made by the French and later, the Swiss.

The American Watch Industry began in 1850. It was started by Aaron Dennison, and is an adaptation from the European method.

For a watch to be considered an antique, it must be at least 100 years old. A vintage watch is any other watch that has not yet reached 100 years

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http://www.studiosoft.it/AntJewelryWatches.htm

Extraordinary Stories behind Ordinary Objects That Had to Be Invented by Someone

Invention: Wristwatch
Inventor: Jacquet-Droz and Leschet, Switzerland
Year: 1790

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How Invented: The first wristwatch was introduced in Geneva, Switzerland, by the watchmakers Jacquet-Droz and Leschet as a watch fixed on a bracelet. During the next century, these bracelet watches remained exclusively items of jewelry for women. The most famous surviving example belonged to the Empress Josephine. Encrusted with pearls and emeralds, this bracelet with a gold watch was made in 1806 by Nitot, a Parisian jeweler. Despite Josephine's leadership, it was not until 1908 that Parisian women made wristwatches fashionable. Among men, war, not fashion, brought the wristwatch into popularity. First in 1880, they were ordered by the German Admiralty for artillery officers, and then in W.W.I, the wristwatch replaced the traditional pocket watch because of its practicality during active service.

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http://www.trivia-library.com/b/story-behind-inventors-and-inventions-wristwatch.htm

The wristwatch, today the most common watch available, was first invented by Patek Phillippe at the end of the nineteenth century. When first invented the wristwatch was worn by women only and it wasn't until the beginning of the twentieth century that the wristwatch was worn by a man, Mr. Alberto Santos-Dumont. Mr. Santos-Dumont, busy working on the invention of the aeroplane, had difficulty checking the time while in his aircraft. He turned to his friend, Mr. Louis Cartier, and asked him to create a watch just for him. Cartier created the perfect watch for Santos-Dumont and soon he was selling wristwatches to other men all around Paris. During World War I, military personnel found that the use of the wristwatch, especially on the battlefield,proved to be far more beneficial than pocket watches. Many European and American officers kept their wristwatches, once the war was over, thus popularising wristwatches in Western civilization!

In 1880 Constant Girard (Girard-Perregaux) develops a concept of wristwatches, made for German naval officers and ordered by Kaiser Wilhelm I of Germany. Two-thousand watches were produced, which represents the first important commercialization of wristwatches.

The wristwatch was invented by Patek Philippe, in 1868, but only as a "lady’s bracelet watch", intended as jewelry. This is the best picture I could find of the bracelet.
http://www.patekmuseum.com/getImage.asp?id=41
In 1904, Alberto Santos-Dumont, an early aviator, asked his friend, a French watchmaker called Louis Cartier, to design a watch that could be useful during his flights. As pocket watches were unsuitable, Louis Cartier created the Santos wristwatch, the first man's wristwatch and the first designed for practical use.