Monowire

Monowire weapons (also called monofilament or mono-molecular, depending on the setting) are a common class of soft sci-fi close-combat weapons. The idea is that, since blades cut better when their edge is thinner, blades one atom thick are the ultimate in slicing tools (short of energy weapons like lightsabers). Some blades of this type may be monomolecular only at the edge, while others are more like a long polymer string or sheet, never more than one atom thick.

Settings with monomolecular weapons may or may not explain how blades one atom or molecule thick manage to hold together for any reasonable length of time. The answer usually involves hand-waving about a stabilizing force field or an exotic or otherwise non-natural molecular bonding. Alternatively, they may explicitly declare that such weapons are extremely delicate, requiring frequent replacement of the blade. Blades that are only monomolecular at the edge are more easily justified: the material may be extremely chip-resistant (in order to be able to be sharpened so much in the first place) or tempered so as to resharpen itself via flaking (i.e. a blow to the edge causes the material to flake, exposing a fresh cutting surface underneath). If the blade is made well enough, it may be so sharp and smooth that only another monoblade can concentrate the energy of a blow enough to damage the edge -- anything duller will simply be sliced cleanly.

Real Life

Obsidian blades can have edges that are only a few hundreds of atoms wide, while the sensor needle of scanning tunneling microscopes tapers down to a single atom at the tip. STM's aren't very useful as weapons, but obsidian was and is a choice material for making extremely sharp arrowheads and knives -- obsidian scalpels can be so thin that they slice right through nerve cells, allowing incisions to be made without requiring anesthesia. Many an archeologist has reached a hand into a cache of artifacts and only realized that there were obsidian blades inside when said hand came out with several fresh cuts.

Some would have you believe that katanas are sub-mono-molecularly sharp (cf. Katanas are Underpowered in d20). They are wrong.

Warhammer 40,000

The Eldar and Dark Eldar of Warhammer 40,000 use guns that shoot monowire nets that slice up infantry.

The edges of Astartes combat blades and Imperial chainsword teeth are said to be mono-molecular as well. How much does the Imperium treasure its mono-molecular blades? A pair of PDF scouts who found an Standard Template Construct printout that detailed how to fashion more durable, cheap mono-molecular blades were awarded with the governorship of a planet, each.