We don't know for sure where the bow originated. There are claims that the bow has been around for over 60,000 years, based on finds of small bone and stone points. Wadley (2008) describes possible stone arrowheads from Sibudu Cave, South Africa, 64,000BP. The stone tools are arrowhead sized (comparison being with North American arrow and dart heads), but IMO they don't look convincingly arrowhead-like in shape. They are possible arrowheads. Backwell (2008) describes possible bone arrowheads, also from Sibudu Cave, 61,000BP. These points look very much like ancient and modern bone arrowheads (but also like bone points on spears). If these are arrowheads, then archery is over 60,000 years old, and probably originated in Africa.

Shea (2006) analyses stone points, and concludes that we don't have good evidence of stone projectile points before about 40,000 years ago. This disagrees with Wadley (2008) (and is discussed therein). However, the projectile-point-like bone points are still excellent candidates for older projectile points, even if Shea is correct. Brooks (2006) notes that Middle Stone Age points are heavier than typical modern stone arrowheads, and suggests that the MSA points are heads of spear-thrower darts. Brooks also suggests that the development of improved projectile technologies might have led to the spread of anatomically modern humans from Africa c. 60,000BP.

Definite evidence of archery turns up much later. We have possible bow fragments, from northern Europe, from about 12,000BP. The oldest bows date to about 10,000BP. Archery is also depicted in rock art from Europe and Africa, many thousands of years old, but quite possibly younger than the oldest known bows. If archery developed not long before this definite evidence, we don't know where or when. Archery spread through Eurasia and Africa early enough so that we don't have good evidence.

The absence of pre-modern bows in Australia suggests that archery might have developed after the settling of Australia.

In summary, we don't know where or when. At least 10,000BP to possibly before 60,000BP. If the earlier dates, the bow was invented in Africa. If a much later date, then somewhere in Africa or Eurasia.

We have much better evidence for the arrival of, and the spread of, archery in the Americas. Archery in the Americas appears to result from spread of the technology from Asia, through Alaska and the Arctic. An oldish, but good, overview of the diffusion of the bow in the Americas is given by Blitz (1988). Possible arrow points appear in the American Arctic 11,000-8,000BP, and clear evidence says the bow was present there by 5,000BP. The bow might only have spread south of the Arctic in the early first millennium AD, and spread throughout North America during the 1st millennium.

References:

Backwell, Lucinda, Francesco d'Errico and Lyn Wadley, 2008, "Middle Stone Age bone tools from the Howiesons Poort layers, Sibudu Cave, South Africa", Journal of Archaeological Science 35(6), 1566-1580.

Blitz, John, 1988, "Adoption of the Bow in Prehistoric North America", North American Archaeologist 9, 123-145. http://anthropology.ua.edu/reprints/22.pdf

Brooks A.S., Nevell L., Yellen J.E., Hartman G. (2006), "Projectile Technologies of the African MSA". In: Hovers E., Kuhn S.L. (eds) Transitions Before the Transition. Interdisciplinary Contributions To Archaeology. Springer, Boston, MA

Shea, John J., "The origins of lithic projectile point technology: evidence from Africa, the Levant, and Europe", Journal of Archaeological Science 33, 823-846.

Wadley, Lyn, 2008, "The Howieson's Poort Industry of Sibudu Cave", South African Archaeological Society Goodwin Series 10, 122-132.