One still often finds books that state "no one knows exactly when
and where the tarot was invented". While technically true, the
statement is quite misleading. The first appearance of the cards can
be dated to within a decade or two, and the location of their origin
can be identified to within a few hundred miles. This is in fact
rather remarkable precision for an item of popular
culture. 
We know that playing cards (the ordinary sort, not tarot) first entered Europe in about 1375. From 1377 onward, there are numerous references to playing cards, in books about games, in city ordinances prohibiting gambling, and in the sermons of clerics opposed to such pastimes. Before 1375, there are no references to cards at all, anywhere in Europe, even in lengthy books that describe all forms of games known at the time. The earliest references to playing cards describe them as "a new game" or "newly arrived". Playing cards apparently entered Europe from the Islamic world, where they were already common and had been known for centuries.
But what about tarot cards? Although there is one intriguing reference from about 1420 to cards with allegorical pictures of animals, it is not until the 1440s that references to triumph cards begin to appear, and such references are found in every decade thereafter. The first is from Ferrara in 1442, describing the acquisition of some triumphs cards for use at the royal court. In 1450, the newly installed Duke of Milan, Francesco Sforza, wrote a letter to one of his underlings requesting the purchase of several packs of triumph cards for use at court on a special occasion. The letter is interesting for several reasons. First, it mentions both triumph cards and playing cards, and clearly distinguishes them. Playing cards were the duke's second choice, to be purchased in case triumph cards were unavailable. It thus provides evidence of a clear distinction between the two types of deck, and incidentally makes it clear that the triumph cards were being used to play card games, not for instructional or esoteric purposes (playing cards would not be a reasonable substitute for those purposes). Secondly, the letter confirms that the triumph cards were being manufactured and sold for general use; they were not strictly expensive art objects commissioned by the royalty, like the few decks surviving from this period are.
So from about 1375 into the early decades of the fifteenth century, ordinary playing cards became familiar objects throughout Europe, and by the 1440s in northern Italy, a second type of cards, triumph cards, had come into use. Since the first mentions from Milan and Ferrara use the phrase "triumph cards" without explanation, it is fair to assume the cards were already generally known in those cities in the 1440s. The absence of any mention of them in earlier documents pertaining to cards, however, means that they could not have been too old at the time of these first references. The most likely date for the invention of the triumph cards is thus the decade of the 1430s, or perhaps a bit earlier. This is consistent with the evidence of the oldest surviving cards themselves, which art historians have dated to the reign of Filippo Maria Visconti, Duke of Milan from 1412 to 1447.
All the surviving triumph cards from the 15th century are from northern Italy. The map shows northern Italy around the end of the century. The region was at this time a patchwork of city-states, often in political or military contention with one another. The church owned much of central Italy, as far north as Bologna. The kingdom of France bordered on Savoy to the west. The major cities where triumph cards were made and used during this early period were Milan, Ferrara, Venice, Bologna, and Florence. By the early 16th century, the cards had spread to France. Around 1530 the word tarocchi (Italian ancestor of the French tarot) first appears. The reason for the change in name is apparently that card players had made an innovation, discovering that the game of triumphs could be played with an ordinary playing-card deck, if the players simply declared a particular suit to serve as "trumps" at the beginning of the hand. (Players of Bridge will find this familiar.) Hence "triumphs" ("trumps") became an ambiguous term, and a new word was needed to refer to the traditional triumphs game with its special picture cards for permanent trumps. Thus the word tarocchi came into use, although its etymology remains a subject of conjecture.
Of course, the historical evidence only documents the birth of the triumph cards as a card game. Is it possible that the symbols we see on the 22 special cards of the tarot deck were first brought together for some other purpose, and only later combined with a deck of playing cards to make a game? Indeed, it is possible. But if that happened, not one piece of corroborating evidence has survived. Although the individual symbols on the cards (the virtues, the Devil, the Sun, the Wheel of Fortune, and so on) are all, almost without exception, found elsewhere in the art of the Italian renaissance, we find not a single example of anything close to this particular sequence of 22 symbols being used together for any purpose until they were placed on playing cards by some inventive Italian game designer early in the 15th century. Still, this does not imply that the symbols were selected arbitrarily. It seems more likely, given the subjects of the cards and their ordering, that the designer was following a deliberate plan, even if it is no longer obvious to us exactly what that plan was.