The invention of the bow and arrow occurred so long ago that it is extremely difficult to determine when and where it happened.
Remains of ancient wooden bows only survive under very specific circumstances – when the wood is either kept consistently very dry (deserts) or very wet without oxygen (in peat bogs or the silt in lakes and occasionally the sea). As the bow was invented tens of thousands of years ago, the massive climatic changes at the end of the last ice age (which raised sea levels and moved deserts, ice sheets and peat bogs), significantly limits how far back we are ever likely to find a bow.
Most of the surviving stone age bows are from northern Europe. This is likely to be related to the presence of stable peat bogs and more numerous archaeological excavations, rather than proving the bow was invented here.
Flint points survive much more easily than the bows, but there are problems using these to date the invention of the bow. Not all arrows would have had a flint point (most arrows collected from the rain forests of Papua New Guinea and Brazil have wooden broadheads or points), and not all flint points are from arrows.
The oldest flint projectile points date back about 50,000 years and have been found in the Tunisian Sahara (which was not a desert then). 
Recently there have been claims that some bone points from South Africa about 60,000 years old were arrow points as they were similar to more recent Bushman bone arrow tips.

The oldest finds which are nearly complete and undisputedly bows are the Mesolithic finds from Holmegaard. These were found in a peat bog in Denmark, and are dated to about 9200 years ago. One and a half bows have been found, of very similar design, one 60" long with limbs about 2" wide, the other about 64" long with 2.4" limbs. They are flat bows of sophisticated design made from small elm trees. They have crowned backs from the under-bark surface of the tree, with flat bellies. The grips are narrowed and deepened. About half way along the limbs, the limbs taper in width to pointed tips without nocks.

The Holmegaard bows

 

 

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                                   Note that the bows do not have an exaggerated step in limb width like a number of the replicas which are mistakenly called Holmegaards.

There are older reported finds of bows, but with less certainty of the identification. The most famous of these is the bow tips from Stellmoor in Germany. These were made from pine, which is generally considered an appalling wood to use for a bow. Recently it has been suggested that the wood was probably pine compression wood (from a tree that had leant over and grown back straight) which is much stronger than normal pine wood and has been used in more recent bows from Lapland. These fragments date from about 11,500 years ago, which is at the end of the last ice age. There were probably no other types of tree growing locally at that time that could have been made into a bow. The identification of the finds as parts of bows was made more likely by the finds of numerous arrow shafts of slightly earlier date at the site.

A find of less certain identification is another pine possible bow tip from Mannheim in Germany and dating to nearly 16,000BC.

Mesolithic Bows

The Mesolithic is a term used by archaeologists to identify the period between the end of the last ice age and the beginning of agriculture. A surprising number of bow from this period have been found, possibly more than 30.

The bows range from simple tapered branches through to sophisticated flatbow designs (like the Holmegaard). The most common wood to use was elm, with odd finds of other woods such as rowan and dogwood. Elm has the reputation of being decay resistant when submerged, so the preponderance of elm bows may be an accident of their survival.

All the bows are made from small section wood. Two inch diameter is plenty for one of these designs and their bowyers would not have wanted to cut down and chop away a large amount of unnecessary wood with stone tools.

The earlier finds had not developed string nocks. The tips were just tapered and the string tied on. This works surprisingly well and the string tend not to slip.

As well as the previously mentioned Holmegaard, Demark hosts another impressive site at Tybrind Vig, a shallow coastal site off Denmark. This has produced a conventional flatbow and at least three large fragments of flatbows with an exaggerated step in the limb width at mid limb.

Neolithic Bows

The Neolithic (new stone age) is the period from the beginning of agriculture to the development and use of bronze (the Bronze Age).

Yew started to dominate the Neolithic bows, probably as the tree started to re-colonise northern Europe after the ice age. Yew is more elastic than the hardwoods, and can therefore be made narrower. This enables a much smaller tree to be used, which is useful if you have to cut it and remove all the waste wood with stone tools. Some bows continue with the Mesolithic flat-bow designs with flat belly and back found from the under-bark surface. Others were early longbows which unlike the medieval bows did not use a sapwood layer for the back, they were shaped from the heartwood of the tree, often from quite small branches.

There are a few notable UK finds, all from yew – the wonderfully named Rotten Bottom bow from Scotland dates to the Neolithic/Mesolithic transition and is a flat-ish longbow.

Two interesting bows have been found near Glastonbury. The Meare Heath bow is the slightly older at 4,690 years old. This is half of a massive yew flatbow, about 2.7" wide and assuming the bow was symmetrical (it broke in the grip), about 75" long. It had a crowned back (not following a growth ring), and a flat belly developing into a keel near the grip. The bow was decorated with a number of leather bands in a criss-cross pattern. It was massively overbuilt for a yew bow, and would have been more efficient if it was much narrower and slightly shorter, but it must have been an impressive sight. The 4,665 year old Ashcott Heath bow is another yew bow, this time closer to a longbow in design. Again it was broken in the handle, and only half survives.
The Meare Heath bow

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Archery was used for warfare in the Neolithic, as a number of hill forts show evidence of sustained archery attack. A large number of flint heads have been recovered from the Crikley Hill fort near Cheltenham, with concentrations of over 400 in the two entrances. Several other hill forts show evidence of archery attack, some with skeletons with embedded flint arrowheads (for instance Hambledon Hill in Dorset, dated to 5,400 years ago).

The UK was not the only place that archery was used for warfare, as at Talhiem in Germany, a 7,000 year old mass grave has been found with several of the 34 victims having arrow wounds. A number of Neolithic cave paintings from Spain show bands of archers shooting at each other.

One of the most televised bows has to be the 5,250 year old yew longbow found with Ötzi the iceman, discovered in a glacier in 1991 on the Italian/Austrian border. This bow is a 73" yew longbow, probably unfinished (it is covered in scallops from his copper axe). This bow is most interesting in that it was found with the archer (allowing comparisons with the archers height of 5'3"), and all his equipment. Recent investigations have found that Ötzi was shot in the back by an arrow, which penetrated his shoulder blade and left the flint head in the body.

Bronze Age

A number of bows have been found in lake and river beds in Italy. The most interesting is the 3,600 year old recurve from Lake Ledro. This is a surprisingly modern looking 57" yew bow. It has very flat, wide thin limbs with recurved tips, and a very narrow and deep handle section. Although it does not have an offset sight window (or fittings for long rod, sights and clicker), it is about as close as you can get to a modern target recurve made from a single piece of wood.
The Lake Ledro recurve

 This information is from the Prehistoric Archery & Atlatl Society with their open permission