Look up the Loadiine browser exploit. Backups from SD only.
Region free (however, DLC and digital games are region locked)
Best bet these days is Sky3DS flash cart. DS games are region free, as long as they don't have any DSi "enhancements" (Pokémon Black2/White 2 for example does not work)
JAMMA standard is international
2600 games are region free (may need vertical hold adjusting on TV), 7800 games often have issues and rarely work: [Link]
Action Replay demo disc (preferred) or Utopia boot disc
All models region free, GBA and GBA SP also fully backwards compatible with all regions
JP/US mod switch (can also use Freeloader disc)
Game Genie works as a pass-through device for MD games. Power Base Converter (model 1 only)/Master System Converter II (models 1 or 2, only sold in Europe, no card slot) allegedly works with any region SMS cart (even though JP SMS games have a wider PCB?), but SG-1000 games won't work on ANY Mega Drive or Genesis.
EU games work full-screen and full-speed on a US system by default, however unlicensed EU cartridges are incompatible. There is a converter to play US games on a JP SMS, but it's very unreliable (only works with a few of the early games). JP carts are too thick for a US system (and the PCBs too wide), and US carts are too wide to fit in a JP console. JP (not US/EU!) games that came on cards are region locked. SG-1000 on a non-JP SMS is impossible.
A US console with the cart slot tabs removed to play JP games is the best option. You can also put the PCB of a Japanese game into the plastic casing of a cheap US game cartridge and use that. I use a PAL N64 with a "passport" converter (there are many versions due to compatibility issues and the latest ones required Action Replay-style codes to work, very inconvenient). There aren't many worthwhile Japanese N64 games anyway.
Region free for DS and GBA, including "DSi enhanced" games.
A US NES or NES2 can play FC games with a HoneyBee passthrough device. I use a generic converter with no outer casing to play US NES games on my Famicom.
Some of my PS1s are modded, also debug PS1s and Net Yarozes are region free by default. The best non-hardware option would be to load your import games via an Action Replay or similar cheat device (this also enables you to play games that detect mod chips, such as Ore no Ryouri).
I don't recommend using a modchip because they seem to strain the laser (heard many tales of PS2s dying shortly after having one installed). I use a silver slim model (SCPH-75003) and have the door sensors blocked using white tack. Swap Magic 3 to play import PS2 games, and Action Replay V2 bonus disc (PS1 codes) to play import PS1 games. Swap Action Replay V2 with PS1 GameShark 4.0 for American consoles. Also, not all debug PS2 units are region free, and some can't run retail versions of games!
Action Replay 3-in-1 (NOT 4-in-1!) handles all imports, including games that require RAM expansion carts. However, Panzer Dragoon Saga requires a code to work - Master Code: F6000914 C305 B6002800 0000 Cheat Code: 06002EB0 0009
JP only system, compatible with SG-1000 II, Mark III and JP SMS
A US system with the cartridge slot tabs removed can play JP games just fine (but for PAL you'll need a converter, HoneyBee seems to offer the best compatibility).
CDs are region free, but for HuCard games you're best off using a US Turbo Duo with a converter (expensive though). Japanese systems (PCEs) require a hardware mod to play US HuCards.
Homebrew channel with Neogamma launcher.
Region free... sort of. Check the compatibility list at [Link]