Flex or flexible nibs are fountain pen nibs that produce a line proportional in width to the pressure used. A very flexible nib can produce a width variation of about six times. Extremely flexible nibs are sometimes known among collectors as "wet noodles".
Flex nibs should not be confused with nibs that merely offer a cushioned feel in writing, or even those that vary line wetness and colour saturation with pressure. Such nibs are known merely as 'soft'.
Such nibs are usually of 14K gold alloy. Alloys with a higher proportion of gold—most commonly 18K—are too soft to flex properly; rather than flexing they are liable to bend and then stay in their new position. Vintage steel nibs can have flex, the best known example being the Esterbrook 9128 nib. However, steel flex nibs on fountain pens are rare; flex nibs required a considerable investment of skill to make, and the steel nibs of the time had poor resistance to corrosion from (often acidic) contemporaneous inks.
Even moderately flexible nibs are not easy to use. Pressure must be carefully controlled to avoid damaging the nib, and as the nib tines spread apart frictional forces with the page can greatly increase. The latter means that pressure and hand motion must be coordinated so that maximum pressure is associated with down strokes that pull the nib in the direction of least resistance.
Flex nibs were much more common on pens made before the 1930s. Typically they were offered as an option on a manufacturers' pens, so that the same model could come with a standard rigid nib or flex. Flex nibs were relatively common on Waterman pens, with the model 22 being particularly associated with them, and particularly rare on Sheaffer pens. Flex nibs remained relatively common on some European pens into the 1950s, notably on Mont Blanc pens.
Flex nibs used for the Spencerian or Copperplate writing styles should possess the property of "springback" or "return," meaning that their tines should close back together extremely quickly when released. This is essential to the rapid thick-to-thin transitions the style requires. 
Flex nibs are commonly available for dip pens, owing to their market and intended use. These are almost always made of steel, because flexible nibs can be made more easily with steel alloys than the available gold alloys.
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Flex nibs, while still widely available in dip pen form, are quite rare on modern fountain pens. This is partly due to the popularity of script styles using flex pens having diminished in the early part of the 20th Century towards more rigid nibs, and partly because of the skill they require to be used without damaging the nib's tines. The Pilot Namiki Falcon is an example of a modern pen with a somewhat flexible nib, although its degree of flexibility is very moderate by vintage standard, reducing the danger of damage and difficulty of use. An even more flexible contemporary pen is the Pilot Custom 742 and 743 with Falcon nib. These pens are much more flexible than a Pilot Falcon (aka Namiki Falcon). A very few "nibmeisters" (or nib-modifiers) can both add flex and grind down the tips of modern 14K nibs to more closely match earlier examples of fountain pen flex nibs. There have also in recent years been several relatively cheap flexible nib fountain pens come onto the market, namely Noodler's Creaper and Ahab models, which use steel-alloy nibs in lieu of 14K gold-alloy nibs to achieve a wide range of flex. These nibs, while often a great introduction into the ability—and art, as most calligraphers would argue—of flexible nibs for new users, they lack some of the control and finesse of gold nibs, and the capability to make hairlines that traditional steel dip nibs possess. These nibs also don't possess the same "spring-back" that some 14K fountain pen nibs offer. They are, however, more forgiving in the accidental case of over-flexing, given steel's more resilient characteristics, and at a price point that's accessible for most people.