(Author's Note: Bows are probably a fairly widely homebrewed resource, because they are given as an example, yet remain unstatted. This entry does not assume any particular stats for bows, it is more just a general commentary on their possible place in the universe. My personal preference for stats is picking a few from AUGC by SquireNed to utilize as appropriate)
The bow and arrow is a tool for hunting and warfare which predates history. A simple device, it's primary use in warfare was long supplanted by the development of gunpowder, but for it's simplicity and history the design and methodology has retained even after The Fall. Archery has a strong cultural connotation for many, both as a sport, an art and even in spirituality - folk heroes, legends and mythological figures can be associated with the bow, or are said to govern the art.
With the wide spread of knowledge and technology, bows remain fairly simple to make. Almost anybody with survival skills can scrounge a basic "self bow" from line and raw materials, while tools or fabbers can produce a basic recurve or composite bow. While they take a little assembly time, compound bows are also fairly easy to produce using carbon composites - and some designs take it further - shape adjusting smart material bows, or self pulling "automatic bows". Arrows have a similar revolution, from chips of glass and volcanic rock on carbon tubing to monofilament accushot smart arrows. The variance of gravity and atmospheric density, combined with overall greater physical aptitude of transhumanity can similarly make very impressive displays of archery.
Archery is preserved as a traditional sport many places, with target archery still being the most common - but Field archery is practiced on Mars, and the largest and greenest Lunar cities and Cylinder habs which have the available space. As a classical art, archery also persists. HEMA enthusiasts on Venus have recovered or engineered several historical styles, including a revival of the English longbow. Traditional archery is popular on Luna, with their high Indian population - the Sanskrit word for "archery" was eventually applied to all martial arts. Hybrid English-Indian systems and styles are popular in the Outer System. The real place to go for the competitive market in archery sports and arts is Mars, however. North African and Central Asian diaspora have preserved horseback archery, even some attempts at chariot archery. There is enough East Asian diaspora that traditional arts like Korean Gungdo and Japanese Kyudo are still taught, while Indian, European and American influences also exist. And Mars has the space and low gravity for some impressive shooting. Sadly, outside of neo-primitives and the exotically rich, bowhunting and fishing has gone almost completely extinct.
But, archery is not just a sport or an art - the bow is a weapon. While their efficiency is still outpaced by more modern weapons, they have advantages which a skilled user can make good use of. Bows are very quiet compared to firearms, seekers and even railguns which still have a supersonic report. They are much less disrupted by atmospheric effects than energy weapons (especially with the right smart arrow), and have a unique advantage that their ammunition can often be recovered after a fight. All this while retaining a decent range, accuracy and damage profile. The Rajput of the Ultimates swear by the effectiveness of a proper bow in the hands of the adequately skilled and augmented in killing "soft" biomorph exsurgents and small TITAN bots - and doing so quietly. Martian Nomads sometimes use them to pick off small Wild Artificials or neogenic creatures like Causapods. They also occasionally use them to quietly dispose of bandits and corporate busybodies who trespass.
The simplicity and construction of some bows means they can even be concealed past some security measures, so they have been occasionally used in assassination attempts, terror attacks and intrusions. Some bioconservative or even primitivist groups have used simple bows to kill as a public statement, where as quiet intrusion specialists just prefer the low sound level to lethality ratio. This also doesn't discount fantastic phantom thieves and hitmen who just like to show off. The Way of the Bow is alive and well in AF 10, in a multitude of ways.