In 1991, PCs predominantly ran on Microsoft's DOS Operating System. Unlike today, where the GUI (Graphical User Interface) is taken for granted in PCs for the OS and applications, the GUI and the OS were separate products. Extremely primitive by today's standards, DOS was a 16-Bit character-mode OS and had no built-in multitasking capabilities. It also used an unjournaled 16-bit directory-based filesystem, FAT, which was used on both floppy and hard disks and had an 11 character limit for filenames, hence the "8.3" file format with names such as AUTOEXEC.BAT.
In addition to Microsoft's DOS, IBM had it's own version, PC-DOS, that ran specifically on its PS/2 personal computers. Digital Research, which pioneered in the late 1970's with the forerunner to DOS, CP/M, also released its own DOS-compatible OS, DR-DOS, and eventually ended being owned by Novell and later, SCO.
Developed by Quarterdeck Office Systems, DESQView was a popular "Shell" for DOS which enabled a primitive form of multi-tasking of applications. This allowed popular productivity applications such as Lotus 1-2-3, WordPerfect and Harvard Graphics to be task-switched. In combination with a special memory manager known as QEMM on Intel 286, 386 and 486-baeed PCs, it allowed users to take advantage of more than 640K of memory and have several applications running in resident memory simultaneously.
DESQView, which was released in July 1985 -- only a few months before the release of Windows 1.0, was not the first task switcher for PC's -- that distinction goes to IBM's TopView, which never particularly caught on. DESQView itself would soon find itself in competition with Windows 386 and in 1990, Windows 3.0, which was a true, although non-preemptive multitasking GUI. By 1992, Quarterdeck did eventually release a full GUI version, of DESQView/X, but by then it was too late -- Microsoft had seized control of both the OS and GUI market for DOS with Windows 3.1 and its Office suite of native applications.
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