PPSSPP can run your PSP games on your PC in full HD resolution, and play them on Android too. It can even upscale textures that would otherwise be too blurry as they were made for the small screen of the original PSP.
Even on modern Android phones and tables, you can often run at double the original resolution.
Compatibility is constantly increasing with each new release. A compatibility list can be found here.
* Play in HD resolutions and more
* Play on a tablet for big-screen mobile gaming
* Save and restore game state anywhere, anytime
* Crank up the anisotropic filtering and texture scaling
* Continue where you left off by transferring saves from your real PSP