ASuW (pronounced "ae-sue-doubleyou") is
milspeak for "Anti Surface Warfare." This means
fighting or attacking targets on the surface, as opposed to in the air (
AAW) or
submerged (
ASW). Typically used in the
Navy (U.S. Navy, at least) community to refer to fighting
surface combatants, or to describe the ability to do so. Units are described as 'having ASuW capability' if they can engage such targets.
Multi-role ships can perform more than one of these roles at a time; the 'single-mission ship' is a rarity in the U.S. Navy these days as lower unit counts and higher requirements mean that ships must be able to perform multiple duties. Modern ASuW vessels are quite rare, mostly due to a technological accident: the weapon of choice for ASuW is the surface-to-surface missile. These weapons, such as the AGM-84 Harpoon and the Exocet sea-skimmers, can be launched from VLS cells, which are also used to launch AAW weapons such as the Standard missile. Therefore, all that is required to have a multi-role ship is additional sensor and targeting modes and equipment. The DDG-51 Arleigh Burke class destroyer is the current U.S. Navy archetype of the multirole ship; future combatants will most likely follow in its footsteps.
Although ASuW may have become less important as naval combat has shifted from close-in gun duels to over-the-horizon missile and aircraft strikes, it is nonetheless what the Navy sees as its 'traditional primary mission' - and hence will never vanish. Even those ships which are tasked with another combat mode as their primary role will practice ASuW any way they can, just to be prepared.