Fewest Moves (or Fewest Moves Challenge, FMC) is an event where competitors attempt to solve a puzzle (almost always the 3x3x3) in as few moves as possible, starting from a given scramble. The usual turn metric is HTM.
There is one WCA-official Fewest Moves event: writing down the shortest possible solution to a 3x3x3 scramble, with a time limit of one hour. Other types of fewest move solving have been tried in the Fewest Moves Challenge, with the most popular other events being the Linear Fewest Moves event (solving the 3x3x3 with as few moves as possible, without being able to undo moves) and the event of writing down the shortest possible solution for the 3x3x3 with a time limit of one week.
The original Fewest Moves Challenge (FMC) was a challenge started by Dan Harris. It was the first online Fewest Moves competition and, although it is no longer run by Dan, it remains a good way to practice fewest moves solving. One scramble is given per event every week, and the results are typically up soon after the next round starts.
Although it is very likely that fewest moves solving was first experimented with in the 1980s, the event was popularized by the, now defunct, website Dan's Cube Station with its Fewest Moves Challenge (FMC), which began in April 2003 and was held nearly every week for many years. The event was also held at many competitions starting with the World Championship 2003, and recently it has become even more popular due to the weekly competitions at the speedsolving.com forum. Since 2009 this competition was taken over By Per K Fredlund on FMC Challenge
The first world record for the WCA event was set in 2003 and was 29 moves by Mirek Goljan. He improved this result to 28 moves in 2005, a record which stood for more than three years until it was broken by Guus Razoux Schultz, 27 moves in 2008. The record then was brought down to 22 moves by Jimmy Coll and István Kocza. After over 3 years, the current record was set at Czech Open 2012 with Tomoaki Okayama presenting a 20 move solution. This was unmatched until Rami Sbahi also presented a 20 move solution at Michigan 2015.
The best known unofficial result (using a properly scrambled cube) is 17 moves in Weekly Competition 2013-15 by Guus Razoux Schultz.
The BH method is a nice complement for the advanced FMC competitor, learning how to deal with optimal commutators helps a lot.