How to run games for DOS

DOSBox (DOS-emulator) is a free and open-source emulator of an Intel x86 personal computer designed for the purpose of running software created for disk operating systems on IBM PC compatibles, primarily DOS video games.
Boxer (macOS) you can launch programs and gameboxes straight from Finder, search your collection in Spotlight, add extra drives in DOS just by drag-and-drop.

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How to run games for Windows

dgVoodoo 2 dgVoodoo 2 is set of old graphics API's for Windows Vista/7/8/10.
nGlide is a 3Dfx Voodoo Glide wrapper. It allows you to play games designed for 3Dfx Glide API without the need for having 3Dfx Voodoo graphics card.
dxWnd is a windows hooker - intercepts system calls to make win32 programs running in a window, to support a better compatibility, to enhance video modes, to stretch timing and emulate CD disk data and audio tracks. It is tipically very useful to run old windows games.
dxWrapper is a .dll file designed to wrap DirectX files to fix compatibility issues in older games.
WineD3D is a DirectX 1-11 to OpenGL wrapper based on WineD3D, which is an almost full implementation of DirectX used in Wine.
Winevdm (otvdm) old Windows programs (including installation programs) can run in 64-bit Windows as if they were ordinary modern Windows programs. More info here.
Wine^HQ is a compatibility layer capable of running Windows applications on several POSIX-compliant operating systems, such as Linux, macOS, & BSD.

Sheep Shaver (PPC) is a MacOS run-time environment that allows you to run classic MacOS application and games.
Basilisk II (68k) is an Open Source 68k Macintosh emulator. It allows you to run 68k MacOS software on your computer, even if you are using a different operating system.

Amiga

WinUAE is an open-source Amiga emulator for Windows.
FS-UAE is an Amiga emulator based on the excellent emulation code from WinUAE.

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Atari ST/STE/TT/Falcon

Hatari is an Atari ST/STE/TT/Falcon emulator for GNU/Linux, BSD, Mac OS X, Windows and other systems which are supported by the SDL library.
Steem SSE is a free Atari STE emulator. It runs almost every ST program ever made without any problems.

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Sega Genesis/Mega Drive

Gens is a Sega Genesis emulator.
Kega Fusion is a closed-source Sega consoles emulator (including Sega Genesis) with high compatibility.

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Atari 8-bit

Altirra an 8-bit Atari computer emulator. Altirra is designed to emulate the actual Atari hardware as closely as possible.

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Commodore C64/C128

Vice is a Commodore computer emulator. The current version emulates the C64, the C64DTV, the C128, the VIC20, practically all PET models, the PLUS4 and the CBM-II (aka C610/C510).
Hoxs64 is a Commodore 64 emulator for Microsoft Windows 10. Hoxs 64 was designed for Windows 10. Its unquestionable advantage is the ease of use and compatibility with the original Commodore C64 computers.