The development of the Bow and Arrow is considerably older in the Old World than in the New World - perhaps 30,000 years old. In the New World it is a relatively late development with (perhaps) early roots. Dates for initial development of this effective technology vary widely and this has been the regularly debated in the New World literature at least for the last 50 years. However, there is no doubt that bow and arrow technology preceded European entry into the New World.

Summary up front: The Bow and Arrow technology comes into use during the prehistoric period at various locations in North America at different times and its presence is seen in some locations as diffusion from adjacent groups and in other locations as independent invention (sometimes multiple times in the same location). The prevailing thought is that while there were multiple origins and in some cases abandonments, the bow and arrow was in use in the Arctic, sub-Artic, and St. Lawrence Basin about 4500 years ago (ya), in southern New England and at least one location in the Great Lakes region by 4000 ya, some small areas of the southeastern woodlands by 2000 ya but most of it by 1000 years ago, the Great Plains by 1800 years ago, the Plateau by 2300 ya, the Southwest and Great Basin by 1400 years ago, and in California and the Northwest Coast at 1300-600 ya.

Archaeologists discriminate between bow and arrow points and [atlatl darts] (http://archserve.id.ucsb.edu/courses/anth/fagan/anth3/Essays/99/f3.jpeg) on the basis of projectile point sizes and other attributes. The recent literature is primarily focused on using morphometric attributes of projectile points, particularly length, width, weight and stem width and various combinations of these measurements. There is general agreement that if the point stem, or hafting element in the case of side notched types, is less than 9 or 10 millimeters in width and the point is thin and thus light in weight, you have an arrowhead. Specimens larger than that, especially with regard to neck width and weight, are thought to be atlatl darts, spear points or bifaces of unknown function. So, in essence, studies have looked for the earliest arrowheads as indicators of the rise of the bow and arrow technology.

The early dates:

The general consensus (and in this specific case there is consensus) is that the first use of bow and arrow technology was associated the the Arctic Small Tool Tradition (Maschner and Mason 2013). The occurrence of small bone points at a number of sites throughout the Arctic and Sub-Arctic during the Archaic Period has led investigators to the conclusion that the small bone points were used on arrows. Some of the bone points were slotted for insertion of micro blades.

Dating the points from the Arctic Small Tool Tradition is made somewhat tenuous because some dates that have been suggested are extremely old. Robert Ackerman working at several sites in central Alaska dated small bone points at multiple sites at 12,000 to 8,000 ya. These points disappear shortly after and do not come back into use until 4,500 ya in that region, then disappear again until 2400 ya. (Maschner and Mason 2013). Most students of Northern North America now accept that 4000- 4500 years ago is accurate (but see Blitz 1988 who thinks it's later). The early Holocene / late Pleistocene dates seem untenable.

Sabo and Early (1988) argue for an early adoption in southeast Texas. Their material dates to around 5000 years ago. Morse and Morse (1983) suggest an 8000 year old date for the emergence of bows and arrows in Arkansas, but it has not gained much traction. Analysis of points in the St. Lawrence basin and southern New England have led authors to posit a 3000- 4500 ya date for those areas. In the Plateau, Ames et. al.(2010) argue that the Cascade Period ushers in the use of bow and arrow beginning as early as 8000 ya and is in full use by 4,400 ya

Ok, those are the early dates and of them, the widely accepted dates are the 4200- 4500 ya date for the far north, and the 3000- 4000 ya dates from the Arctic through the St. Lawrence Basin to southern New England and perhaps the Great Lakes region. The other claims have not been dismissed but they don't have a large following either.

The later dates:

Studies beginning in the late 1970s and continuing through the late 1990's attempted to systematically approach the problem by measuring specific point attributes in large samples of projectile points from large numbers of sites following the pioneering work of Thomas (1978, 1981). The result has been a series of dated projectile point assemblages thought to be the earliest in regions across the U.S. and further north.

Nassaney and Pyle (1999) have summarized the data on early bow and arrow adoption from the Eastern to Midwest US as follows:

Plains - 1850-1750 ya

SE Texas - 1500 ya

Midwest - 500- 4000(?) ya

Northeast 3000- 4000 ya

Southeast 1200- 2000 ya

They suggest analysis of the data from the eastern U.S. "... indicates selective adoption of the bow as early as the Late Archaic period (ca. 2000 B.C.) followed by a lengthy period of experimentation and refinement. It was not until the Late Woodland period (ca. A.D. 700) that widespread adoption occurred throughout the East." The compelling data supporting early emergence (Late Archaic) appears to be limited to the Northeast and in the northern Midwest although there are hints of it also from Arkansas.

There have been a number of debates over this issue beginning in the late 1970s in the Southwest, Great Basin and California. The current thinking is that the bow and arrow originates in this area with the small triangular, side notched and stemmed points at about 1,350 ya (Thomas 1978, 1981; Bettinger 1999).

Although, as noted above, Ames et. al. (2010) argues for very early use of the bow and arrow in the Plateau Region most of the students of Plateau prehistory would place it much later, around 2000 ya. Analysis of large samples from a number of sites salvaged along the Columbia River suggests that the hallmark of the bow and arrow technology in the Plateau is the rise in use of the Rabbit Island stemmed point.

The use of bow and arrow in California and the Northwest coast is generally thought to be later, perhaps 1,300 to 600 years ago (Bettinger 2015). Some suggest that adoption in the far west may coincide with the movement of Athabascans into California.

The data suggests that following a relatively early development in the far North, perhaps 4500 ya, the bow and arrow may have diffused south and to the east to southern New England and the Great Lakes Region. Initial use and development throughout the remainder of the continent is widely variable by region, but large scale adoption falls nicely into a 1500 year period beginning a little over 2000 ya. Whether the spread of use of the bow and arrow is attributable to diffusion or independent invention is contested, but the prevailing thought is that it probably was a bit of both. In some areas it appears to have been invented, then abandoned. Then following a significant period of time, it was reinvented later in the same region.

There are inherent problems with inferring technological invention from point styles. But such is the nature of archaeological inquiry. It should be noted that there was considerable variation within regions and that bow and arrow and atlatl use frequently overlapped for hundreds if not thousands of years - further complicating issues.

Note that I did not include Mesoamerica in this discussion. I leave it to specialists from that area fill in that gap.

References:

Dart and Arrow Points on the Columbia Plateau of Western North America K. Ames, K. Full and S. Davis American Antiquity 75(2), 2010

Orderly Anarchy: Sociopolitical Evolution in Aboriginal California Robert L. Bettinger Univ of California Press, 2015

Point Typologies, Cultural Transmission, and the Spread of Bow and Arrow Technology in the Prehistoric Great Basin Robert L. Bettinger and Jelmer Eerkins American Antiquity 64(2), 1999

The Bow and Arrow in Northern North America Herbert Maschner and Owen Mason Evolutionary Anthropology Issues, News, and Reviews 2013; 22(3):133-8.

David Hurst Thomas 1978 Arrowheads and Atlatl Darts: How the Stones got the Shaft. American Antiquity, 43(2)

1981 How to Classify the Projectile Points from Monitor Vally, Nevada Journal of California and Great Basin Anthropology, 1981; 3:7-43

The Adoption of the Bow and Arrow in Eastern North America: A View from Central Arkansas. Michael Nassaney and Kendra Pyle American Antiquity 1999; 64(2).

P. A. Morse and D.F. Morse Archaeology of the Central Mississippi Valley, Academic Press, New York. 1983

G.Sabo III and A. Early Prehistoric Culture History, in Human Adaptation in the Ozark and Ouachita Mountains (Sabo et. al.), Arkansas Archaeological Survey, Fayetteville. 1988

John Blitz Adoption of the Bow in Prehistoric North America North American Archaeologist 9(2), 1988

Thanks to /u/Reedstilt for review of an earlier draft of this post.

Edit: Formatting

The bow and arrow definitely predates the arrival of Europeans to North America. Here's a [very brief and rough] timeline of the appearance of the bow and arrow across the northwestern US:

• The Arctic: The first evidence for bow/arrow use is from Alaska ca. 12,000 years ago^1 .

• The Plains: The bow and arrow came into use on the Central Plains and Middle Missouri Valley in the 1st millennium AD^2 .

• The West: The bow and arrow were in use by ca. AD 500^2 .

• The Great Basin: around AD 500^2 .

 

1. Herbert Maschner and Owen K. Mason. 2013. The Bow and Arrow in Northern North America.

2. Brian M. Fagan. 2005. Ancient North America.

Followup question: what are the differences between bows used in medieval Europe and bows used by native Americans? How did the design of this weapon evolve over millennia on both continents?

But wait, there's more! In the FAQ:

• Origins and dispersal of bow & arrow technology