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In mountaineering and climbing, a first ascent (abbreviated to FA in guide books), is the first successful documented climb to the top of a mountain peak or the top of a particular climbing route. Early 20th-century mountaineers and climbers were mainly focused on reaching the tops of notable mountain peaks (e.g. the eight-thousanders such as Mount Everest) and the tops of iconic climbing routes (e.g. the great north faces of the Alps such as the Eiger) by whatever means possible, and often using considerable amounts of aid climbing and/or with large expedition style support teams allowing them to "lay siege" to the climb.
As all the key tops were summited, the manner or "style" in which each top was reached became important to climbers, and particularly the ability to complete the ascent without the use of any artificial aids, which is called free climbing. In free-climbing, the term first free ascent (abbreviated FFA) denotes where a mountain or climbing route is ascended without any artificial aid — note that equipment for protection in the event of a fall can be used as long as it did not aid in the climber's upward progression. Completing the FFA of a climbing route is often called freeing (or more latterly sending) a route, and is highly coveted.
As the sport of climbing developed, additional types of ascent became notable and chronicled in guidebooks and climbing journals. In mountaineering, and in the sub-discipline of alpine climbing in particular, the first winter ascent is also recorded, given the significantly greater difficulty of the undertaking. The first solo ascent is also typically noted, although the first free solo ascent is a more controversial aspect, given the concerns about advocating such a dangerous form of climbing. With the significant rise in female participation in all forms of climbing, the first female free ascent (or FFFA) has also become notable.
As mountaineering developed in the 20th century, the attainment of a summit by almost any means was replaced by ascents that reflected the style used and the conditions faced. In 2008, the most prestigious annual prize in mountaineering, the Piolet d'Or, amended its focus to small light-weight alpine-style teams using no form of aid or support, rather than on large expedition-style teams using "siege" techniques.
The most notable types of mountaineering first ascents that are chronicled are:

In rock climbing, how the first free ascent was achieved became important to chronicle by journals and magazines. The key differentiators were the format in which the route was free climbed (e.g. traditional climbing, sport climbing, or free solo climbing), whether the free climb was done on the first attempt (e.g. onsighted), and whether the climber had prior information (e.g. beta) on that first attempt.
The most notable types of rock-climbing first ascents that are chronicled are:

First female free ascent (abbreviated FFFA). Important female climbers emerged in the 1980s including Lynn Hill and Catherine Destivelle and climbing journals began to chronicle milestones in female first free ascents. In some cases, an FFFA is also an FFA, the most famous example being Lynn Hill's 1993 FFA/FFFA of the 3,000-foot Nose on El Capitan (5.14a/b), considered the biggest prize in big wall climbing, announced by Hill to the world's climbing media by: "It goes, boys". By the 2000s, Josune Bereziartu, Angela Eiter, and Ashima Shiraishi closed the gap to the highest sport climbing grades achieved by men to within one/two notches, while Beth Rodden had fully closed the gap for traditional climbing.
There have been notable disputes over claims of a first ascent (or first free ascent), for various reasons (disputes over the style employed, issues with verifiability, accusations of bad faith and fraud), and the most notable are where a new grade milestone or major advancement in difficulty is being proposed:
The Piolets d'Or (Golden Ice Axes) were long considered to be the 'Oscars of Mountaineering'
A month later, Roxanna Brock teamed up with Bobbi Bensman to do the first female free ascent (FFFA) of the route in a day.