Not to discourage new answers but you may want to see this post here with an answer from /u/Valkine:
/u/AlfredHaZe has already linked to an excellent answer by /u/Valkine, but I thought I'd add some thoughts of my own.
Definitions and Previous Scholarship
I'll start with the question of shortbows, crossbows and longbows. The traditional consensus, down to the late 1970s, was that shortbows were the dominant form of bow during most of the Middle Ages, with the longbow emerging in Britain during the late 13th and early 14th centuries. Whether this was an English development or if they adopted it from the Welsh was occasionally disputed, with the Welsh origin most commonly accepted, but it was generally agreed that the longbow emerged into prominence in the late 13th century under Edward I.
Then along came Robert Hardy and his ground-breaking book (Longbow: A Social and Military History) in 1976. He argued, based on archaeological evidence extended from the Mesolithic to the 7th Century CE, that the longbow (implicitly any bow over 5 feet/152cm) had always existed in some form or other. He further argued that, although the Welsh may have been particularly good archers, if the longbow hadn't been in England before the 9th century, it had certainly come there with the Vikings during their various invasions. This view was rapidly accepted. By 2005, with the addition of the 10th century bows from Ballinderry and Hedeby (each over 190cm long), the case that the "short" bow had never existed was fully entrenched.
However, in 2011 and 2012, two works appeared that challenged this orthodoxy, namely Clifford J. Roger's article "The development of the longbow in late medieval England and ‘technological determinism’" and Richard Wadge's book Archery in Medieval England: Who Were the Bowmen of Crecy?. Both made use of similar material, especially a complete bow (c.126cm long) from the first half of the 13th century that was found with a complete bodkin arrow and a 1297 inquiry into the death of Simon Skeffington, to argue that "short" bows of less than 5 feet had been common in 13th century England. Rogers, using artistic evidence, made the additional argument that the "short" bow was the predominant bow in Western Europe well into the 14th century, while Wadge used the socket diameters of arrowheads from the 12th and 13th century to show that, even if longbows had been in use, they were not much more powerful than an ordinary hunting bow.
Academic consensus has yet to be reached, although John France has recently tentatively accepted part of Rogers' argument on the artistic evidence, but as two additional "short" bows have been found in France and Germany from this period, in addition to two large fragments from Ireland and several smaller fragments from France that are likely to be from "short" bows, it seems likely that many archers did use shortbows into the fourteenth century.
Regarding crossbows, the overwhelming view has been they were powerful, slow shooting weapons and that the trade off between bows and crossbows was power vs speed. Even in the 14th and 15th centuries, when authors see the longbow at its most effective, the argument has been that windlasses and other portable spanning devices allow the crossbow to maintain its edge in terms of power over the bow.
This isn't without primary source evidence. As early as the 11th century, bishop Guy of Amiens indicated that crossbows could shoot through shields, and this was echoed in the early 12th century by Anna Komnena who claimed they could shoot straight through a man, shield and armour included! The widespread use of small numbers of crossbowmen as apparently the sole missile troops throughout the 12th and 13th centuries, combined with continued descriptions of the power and their role everywhere but in English armies, led historians to keep this distinction.
More recently, however, this view has come under more scrutiny. David Bachrach's survey of English crossbow production in the 13th century found that, rather than the old view that composite laths (prods) completely replaced wooden laths going into the 13th century, wooden lathed crossbows remained extremely common and even predominated towards the end of the 13th century. The inventories of the Dukes of Burgundy show that, in the 14th and 15th centuries, this trend was present there as well, and there are certainly some pieces of evidence from French financial accounts that suggest the same trend with regards to the armament of sailors. Although mercenaries, especially the Italians, likely gravitated towards the more expensive composite lathed crossbows, it seems that garrison soldiers, armed sailors and likely also members of the town militias were often armed with wooden crossbows.
The last two decades have also seen many replicas of medieval crossbows made, and some of these have been tested. The most common type, with steel laths, perform noticeably poorly, with a test by Mike Loades showing that the bolts were military ineffective beyond 80 yards. Andreas Bichler's tests with a 289lb composite lath and a 276lb wooden lath, only slightly below the 300lbs of the crossbow used by Loades, appear to offer much better performance, shooting out beyond 220 metres (on par with Mary Rose replicas), but the energy of the bolts is still significantly lower than that of later medieval longbows (55-60j for the wood and 65-70j for the composite).
It's also more generally accepted now that the light crossbows, spanned with a belt and hook and under 300lbs draw weight, were significantly more common than the previous generations of scholars believed and likely were the main sort of crossbow used in the field. Sources such as Christine de Pizan's The Book of Deeds of Arms and of Chivalry, city and royal accounts and manuscript illuminations back up this view.
Now we come to "long" bows. It's important to remember that, even if the longbow was the most common form of bow before the 14th century, not all longbows are equal. A number of 14th and 15th century sources, ranging from murder inquiries to hunting manuals, suggest that civilian longbows were about 70" (178cm) between the nocks and had a draw length of 28" (71cm), which makes them a good deal shorter than the Mary Rose bows and with a shorter draw length. Almost 95% of the Mary Rose bows had an extra 100mm between the nocks, and three quarters of the arrows were sized for a draw length of 30.5".
This information is especially important in the 14th century, when large proportions of Edward III's armies were made up of levied archers rather than recruited professionals. As mentioned earlier, Richard Wadge has found that the socket diameters of 13th century arrowheads indicate that civilian bows were, on the whole, relatively light and probably drew less than 80lbs. Arrowheads from manors, castles and other military sites do show evidence of higher draw weights, and professional archers likely drew bows of 100-130lbs, but many of England's successes in the first half of the 14th century were won by archers who are unlikely to have drawn bows above 100lbs on the whole.
While this does mean that, in theory, all but the weakest longbows had significantly more energy (I estimate that a well made 80lb longbow will put 70j of energy into an arrow) than the types of crossbows commonly in use and much more energy than the typical shortbow, the crossbows are going to be much more consistent in their performance. Bowyers, to judge from 15th century rates of pay for specific parts of the process, were making at least a hundred bows each week in every workshop, and the Mary Rose bows have been noted for their relatively poor finish. While the poor finish doesn't necessarily mean poor performance, 16th century sources say that livery bows performed worse than those made for civilian use and the serial production of the bows makes the kind of care and attention that modern bowyers give to a bow unlikely. Factor in that the arrows were stored and transported in crates, not always of the best wood and were usually made rapidly to fulfil large orders, and it's likely that medieval longbows performed to a lower standard than modern approximations.
(cont.)