The opposite of synchronous. A mode of SCSI operation where each byte is requested, sent, and acknowledged before the next is requested, sent etc. In synchronous SCSI transfers the byte does not have to be acknowledged before the next byte is sent, it only needs to be eventually acknowledged. Most modern SCSI drives can be set to allow synchronous transfers or not. Most SCSI devices can work synchronously or asynchronously, but some systems or SCSI controllers require one method or the other in order to perform correctly (Digidesign equipment, for example, prefers to have synchronous transfers enabled).
Asynchronous also refers to some system commands in the Macintosh, such as copy and format, which take place in the background without tying up the CPU (on new versions of the OS).